Properties

Label 16-637e8-1.1-c1e8-0-8
Degree $16$
Conductor $2.711\times 10^{22}$
Sign $1$
Analytic cond. $448056.$
Root an. cond. $2.25532$
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  − 2·2-s + 3·4-s + 2·8-s + 2·9-s − 4·11-s − 3·16-s − 4·18-s + 8·22-s − 12·23-s + 7·25-s + 2·29-s + 12·32-s + 6·36-s − 20·37-s + 14·43-s − 12·44-s + 24·46-s − 14·50-s − 24·53-s − 4·58-s − 2·64-s + 8·67-s − 16·71-s + 4·72-s + 40·74-s + 12·79-s − 27·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.707·8-s + 2/3·9-s − 1.20·11-s − 3/4·16-s − 0.942·18-s + 1.70·22-s − 2.50·23-s + 7/5·25-s + 0.371·29-s + 2.12·32-s + 36-s − 3.28·37-s + 2.13·43-s − 1.80·44-s + 3.53·46-s − 1.97·50-s − 3.29·53-s − 0.525·58-s − 1/4·64-s + 0.977·67-s − 1.89·71-s + 0.471·72-s + 4.64·74-s + 1.35·79-s − 3·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(7^{16} \cdot 13^{8}\)
Sign: $1$
Analytic conductor: \(448056.\)
Root analytic conductor: \(2.25532\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{637} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 7^{16} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.362493552\)
\(L(\frac12)\) \(\approx\) \(2.362493552\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 + p T^{4} + p^{4} T^{8} \)
good2 \( ( 1 + T - 3 T^{3} - 5 T^{4} - 3 p T^{5} + p^{3} T^{7} + p^{4} T^{8} )^{2} \)
3 \( ( 1 - T^{2} + 5 p T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \)
5 \( 1 - 7 T^{2} - 2 p T^{4} - 63 T^{6} + 1631 T^{8} - 63 p^{2} T^{10} - 2 p^{5} T^{12} - 7 p^{6} T^{14} + p^{8} T^{16} \)
11 \( ( 1 + T - 7 T^{2} + p T^{3} + p^{2} T^{4} )^{4} \)
17 \( 1 - 16 T^{2} + 251 T^{4} + 9168 T^{6} - 157480 T^{8} + 9168 p^{2} T^{10} + 251 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \)
19 \( ( 1 + 63 T^{2} + 1711 T^{4} + 63 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
23 \( ( 1 + 6 T - 6 T^{2} - 24 T^{3} + 407 T^{4} - 24 p T^{5} - 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
29 \( ( 1 - T - 54 T^{2} + 3 T^{3} + 2155 T^{4} + 3 p T^{5} - 54 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \)
31 \( 1 - 72 T^{2} + 2603 T^{4} - 47448 T^{6} + 756216 T^{8} - 47448 p^{2} T^{10} + 2603 p^{4} T^{12} - 72 p^{6} T^{14} + p^{8} T^{16} \)
37 \( ( 1 + 10 T + 14 T^{2} + 120 T^{3} + 2327 T^{4} + 120 p T^{5} + 14 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
41 \( 1 - 8 T^{2} - 2846 T^{4} + 3616 T^{6} + 5534755 T^{8} + 3616 p^{2} T^{10} - 2846 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
43 \( ( 1 - 7 T + 32 T^{2} + 483 T^{3} - 3667 T^{4} + 483 p T^{5} + 32 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
47 \( 1 - 32 T^{2} + 2083 T^{4} + 175264 T^{6} - 6020216 T^{8} + 175264 p^{2} T^{10} + 2083 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \)
53 \( ( 1 + 12 T + 15 T^{2} + 276 T^{3} + 6200 T^{4} + 276 p T^{5} + 15 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
59 \( 1 - 184 T^{2} + 18755 T^{4} - 1497576 T^{6} + 98070104 T^{8} - 1497576 p^{2} T^{10} + 18755 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \)
61 \( ( 1 + 192 T^{2} + 16606 T^{4} + 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
67 \( ( 1 - T + p T^{2} )^{8} \)
71 \( ( 1 + 4 T - 55 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \)
73 \( 1 - 240 T^{2} + 32594 T^{4} - 3443520 T^{6} + 289951395 T^{8} - 3443520 p^{2} T^{10} + 32594 p^{4} T^{12} - 240 p^{6} T^{14} + p^{8} T^{16} \)
79 \( ( 1 - 6 T - 118 T^{2} + 24 T^{3} + 14631 T^{4} + 24 p T^{5} - 118 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \)
83 \( ( 1 + 280 T^{2} + 33053 T^{4} + 280 p^{2} T^{6} + p^{4} T^{8} )^{2} \)
89 \( 1 - 239 T^{2} + 27262 T^{4} - 3350063 T^{6} + 376362199 T^{8} - 3350063 p^{2} T^{10} + 27262 p^{4} T^{12} - 239 p^{6} T^{14} + p^{8} T^{16} \)
97 \( 1 - 141 T^{2} + 74 T^{4} - 139449 T^{6} + 116727639 T^{8} - 139449 p^{2} T^{10} + 74 p^{4} T^{12} - 141 p^{6} T^{14} + p^{8} T^{16} \)
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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

−4.48289715320759449117064495917, −4.39952828842504443741813713425, −4.37406414739981818973190370366, −4.23556784621099796319445078801, −4.08165048108663453776057676734, −4.04365410577749753438400518967, −3.77814706079153294699345162486, −3.62769472018907185756349743647, −3.51236619596134889130721376425, −3.12054763696221523421398594918, −3.09727931737208258718285322023, −2.96290337409701181636073737890, −2.83040553746038624344056564117, −2.76829786986628938002452727435, −2.56039916947045974179445163339, −2.15573981927522526459342177190, −2.02734045614282832183783224516, −2.00406772008254332923907426509, −1.62399163389801151015124751480, −1.51556906462447486507820816520, −1.45489786088699784137537782638, −1.43321711011207616419629326184, −0.821986637802263334066506000391, −0.59491316310844949994328576582, −0.35232875820824479073730201156, 0.35232875820824479073730201156, 0.59491316310844949994328576582, 0.821986637802263334066506000391, 1.43321711011207616419629326184, 1.45489786088699784137537782638, 1.51556906462447486507820816520, 1.62399163389801151015124751480, 2.00406772008254332923907426509, 2.02734045614282832183783224516, 2.15573981927522526459342177190, 2.56039916947045974179445163339, 2.76829786986628938002452727435, 2.83040553746038624344056564117, 2.96290337409701181636073737890, 3.09727931737208258718285322023, 3.12054763696221523421398594918, 3.51236619596134889130721376425, 3.62769472018907185756349743647, 3.77814706079153294699345162486, 4.04365410577749753438400518967, 4.08165048108663453776057676734, 4.23556784621099796319445078801, 4.37406414739981818973190370366, 4.39952828842504443741813713425, 4.48289715320759449117064495917

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

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