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Properties

Label 16-990e8-1.1-c1e8-0-15
Degree $16$
Conductor $9.227\times 10^{23}$
Sign $1$
Analytic cond. $1.52510\times 10^{7}$
Root an. cond. $2.81161$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·5-s + 8·7-s + 8·13-s − 2·16-s − 8·17-s + 16·19-s − 8·23-s + 40·25-s + 16·29-s + 64·35-s + 16·37-s + 16·43-s + 8·47-s + 32·49-s + 24·53-s + 64·65-s − 24·67-s − 32·71-s + 32·73-s − 32·79-s − 16·80-s + 16·83-s − 64·85-s + 64·91-s + 128·95-s − 16·103-s + 8·107-s + ⋯
L(s)  = 1  + 3.57·5-s + 3.02·7-s + 2.21·13-s − 1/2·16-s − 1.94·17-s + 3.67·19-s − 1.66·23-s + 8·25-s + 2.97·29-s + 10.8·35-s + 2.63·37-s + 2.43·43-s + 1.16·47-s + 32/7·49-s + 3.29·53-s + 7.93·65-s − 2.93·67-s − 3.79·71-s + 3.74·73-s − 3.60·79-s − 1.78·80-s + 1.75·83-s − 6.94·85-s + 6.70·91-s + 13.1·95-s − 1.57·103-s + 0.773·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8}\)
Sign: $1$
Analytic conductor: \(1.52510\times 10^{7}\)
Root analytic conductor: \(2.81161\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(68.38318291\)
\(L(\frac12)\) \(\approx\) \(68.38318291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + T^{4} )^{2} \)
3 \( 1 \)
5 \( 1 - 8 T + 24 T^{2} - 32 T^{3} + 32 T^{4} - 32 p T^{5} + 24 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \)
11 \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \)
good7 \( 1 - 8 T + 32 T^{2} - 104 T^{3} + 320 T^{4} - 984 T^{5} + 3040 T^{6} - 9656 T^{7} + 28354 T^{8} - 9656 p T^{9} + 3040 p^{2} T^{10} - 984 p^{3} T^{11} + 320 p^{4} T^{12} - 104 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
13 \( 1 - 8 T + 32 T^{2} - 56 T^{3} + 128 T^{4} - 216 T^{5} - 800 T^{6} + 19288 T^{7} - 88286 T^{8} + 19288 p T^{9} - 800 p^{2} T^{10} - 216 p^{3} T^{11} + 128 p^{4} T^{12} - 56 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
17 \( 1 + 8 T + 32 T^{2} + 88 T^{3} - 132 T^{4} - 568 T^{5} + 3552 T^{6} + 47320 T^{7} + 318854 T^{8} + 47320 p T^{9} + 3552 p^{2} T^{10} - 568 p^{3} T^{11} - 132 p^{4} T^{12} + 88 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
19 \( ( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \)
23 \( 1 + 8 T + 32 T^{2} + 232 T^{3} + 576 T^{4} - 3112 T^{5} - 16416 T^{6} - 141384 T^{7} - 1119806 T^{8} - 141384 p T^{9} - 16416 p^{2} T^{10} - 3112 p^{3} T^{11} + 576 p^{4} T^{12} + 232 p^{5} T^{13} + 32 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \)
29 \( ( 1 - 8 T + 76 T^{2} - 472 T^{3} + 2710 T^{4} - 472 p T^{5} + 76 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( ( 1 + 92 T^{2} + 4006 T^{4} + 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
37 \( ( 1 - 8 T + 32 T^{2} - 328 T^{3} + 3346 T^{4} - 328 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 192 T^{2} + 18436 T^{4} - 1182272 T^{6} + 55788614 T^{8} - 1182272 p^{2} T^{10} + 18436 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \)
43 \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + 2468 T^{4} + 2640 T^{5} - 121472 T^{6} + 1568624 T^{7} - 12738650 T^{8} + 1568624 p T^{9} - 121472 p^{2} T^{10} + 2640 p^{3} T^{11} + 2468 p^{4} T^{12} - 16 p^{6} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
47 \( 1 - 8 T + 32 T^{2} + 328 T^{3} - 1024 T^{4} - 18760 T^{5} + 236640 T^{6} - 189816 T^{7} - 2979326 T^{8} - 189816 p T^{9} + 236640 p^{2} T^{10} - 18760 p^{3} T^{11} - 1024 p^{4} T^{12} + 328 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \)
53 \( 1 - 24 T + 288 T^{2} - 2984 T^{3} + 30976 T^{4} - 278984 T^{5} + 2226656 T^{6} - 17842360 T^{7} + 137588706 T^{8} - 17842360 p T^{9} + 2226656 p^{2} T^{10} - 278984 p^{3} T^{11} + 30976 p^{4} T^{12} - 2984 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \)
59 \( 1 - 216 T^{2} + 24188 T^{4} - 1817768 T^{6} + 113047398 T^{8} - 1817768 p^{2} T^{10} + 24188 p^{4} T^{12} - 216 p^{6} T^{14} + p^{8} T^{16} \)
61 \( 1 - 96 T^{2} + 6848 T^{4} - 210720 T^{6} + 10543138 T^{8} - 210720 p^{2} T^{10} + 6848 p^{4} T^{12} - 96 p^{6} T^{14} + p^{8} T^{16} \)
67 \( 1 + 24 T + 288 T^{2} + 3224 T^{3} + 40988 T^{4} + 439064 T^{5} + 3930080 T^{6} + 36078296 T^{7} + 319412070 T^{8} + 36078296 p T^{9} + 3930080 p^{2} T^{10} + 439064 p^{3} T^{11} + 40988 p^{4} T^{12} + 3224 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \)
71 \( ( 1 + 16 T + 200 T^{2} + 2328 T^{3} + 23168 T^{4} + 2328 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
73 \( 1 - 32 T + 512 T^{2} - 6944 T^{3} + 93188 T^{4} - 14688 p T^{5} + 10708480 T^{6} - 105036128 T^{7} + 966015814 T^{8} - 105036128 p T^{9} + 10708480 p^{2} T^{10} - 14688 p^{4} T^{11} + 93188 p^{4} T^{12} - 6944 p^{5} T^{13} + 512 p^{6} T^{14} - 32 p^{7} T^{15} + p^{8} T^{16} \)
79 \( ( 1 + 16 T + 368 T^{2} + 3720 T^{3} + 576 p T^{4} + 3720 p T^{5} + 368 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( 1 - 16 T + 128 T^{2} - 1136 T^{3} + 6564 T^{4} + 31376 T^{5} - 696960 T^{6} + 8088432 T^{7} - 78605210 T^{8} + 8088432 p T^{9} - 696960 p^{2} T^{10} + 31376 p^{3} T^{11} + 6564 p^{4} T^{12} - 1136 p^{5} T^{13} + 128 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \)
89 \( 1 - 440 T^{2} + 95836 T^{4} - 13716360 T^{6} + 1418326406 T^{8} - 13716360 p^{2} T^{10} + 95836 p^{4} T^{12} - 440 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 + 512 T^{3} - 2428 T^{4} - 67072 T^{5} + 131072 T^{6} - 2606080 T^{7} - 61009914 T^{8} - 2606080 p T^{9} + 131072 p^{2} T^{10} - 67072 p^{3} T^{11} - 2428 p^{4} T^{12} + 512 p^{5} T^{13} + p^{8} T^{16} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.27495229416765950526694596063, −4.21866744624929983105491435468, −4.16102441495696118209952682902, −4.06832583260458091002106538591, −3.77101221269540663301252597909, −3.54524875308051296523921252550, −3.46829591661015962982138562343, −3.30890264686838154513157015080, −3.28601727600077820705644710371, −2.75796641704463493094393953418, −2.71748902700614634459923077593, −2.70821177870632969896556168755, −2.64921383908785079156868217434, −2.36669394919289171609674658491, −2.25856786835245816013874490600, −2.19198980821448204905057011941, −2.07480678968517181755808543785, −1.87188574667195958679435849860, −1.51052737622203350367999591850, −1.26241946801544555425619347490, −1.20410510298558112424101946804, −1.18850809969901523012149970365, −1.11191150602248797037863157240, −0.975042705947011458359065949049, −0.52921979560367787660044569567, 0.52921979560367787660044569567, 0.975042705947011458359065949049, 1.11191150602248797037863157240, 1.18850809969901523012149970365, 1.20410510298558112424101946804, 1.26241946801544555425619347490, 1.51052737622203350367999591850, 1.87188574667195958679435849860, 2.07480678968517181755808543785, 2.19198980821448204905057011941, 2.25856786835245816013874490600, 2.36669394919289171609674658491, 2.64921383908785079156868217434, 2.70821177870632969896556168755, 2.71748902700614634459923077593, 2.75796641704463493094393953418, 3.28601727600077820705644710371, 3.30890264686838154513157015080, 3.46829591661015962982138562343, 3.54524875308051296523921252550, 3.77101221269540663301252597909, 4.06832583260458091002106538591, 4.16102441495696118209952682902, 4.21866744624929983105491435468, 4.27495229416765950526694596063

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.