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Properties

Label 8-546e4-1.1-c3e4-0-1
Degree $8$
Conductor $88873149456$
Sign $1$
Analytic cond. $1.07704\times 10^{6}$
Root an. cond. $5.67582$
Motivic weight $3$
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  + 8·2-s + 12·3-s + 40·4-s + 10·5-s + 96·6-s + 28·7-s + 160·8-s + 90·9-s + 80·10-s + 67·11-s + 480·12-s + 52·13-s + 224·14-s + 120·15-s + 560·16-s + 65·17-s + 720·18-s + 24·19-s + 400·20-s + 336·21-s + 536·22-s + 24·23-s + 1.92e3·24-s − 93·25-s + 416·26-s + 540·27-s + 1.12e3·28-s + ⋯
L(s)  = 1  + 2.82·2-s + 2.30·3-s + 5·4-s + 0.894·5-s + 6.53·6-s + 1.51·7-s + 7.07·8-s + 10/3·9-s + 2.52·10-s + 1.83·11-s + 11.5·12-s + 1.10·13-s + 4.27·14-s + 2.06·15-s + 35/4·16-s + 0.927·17-s + 9.42·18-s + 0.289·19-s + 4.47·20-s + 3.49·21-s + 5.19·22-s + 0.217·23-s + 16.3·24-s − 0.743·25-s + 3.13·26-s + 3.84·27-s + 7.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(1.07704\times 10^{6}\)
Root analytic conductor: \(5.67582\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )\)

Particular Values

\(L(2)\) \(\approx\) \(382.2996057\)
\(L(\frac12)\) \(\approx\) \(382.2996057\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - p T )^{4} \)
3$C_1$ \( ( 1 - p T )^{4} \)
7$C_1$ \( ( 1 - p T )^{4} \)
13$C_1$ \( ( 1 - p T )^{4} \)
good5$C_2 \wr S_4$ \( 1 - 2 p T + 193 T^{2} - 738 T^{3} + 10196 T^{4} - 738 p^{3} T^{5} + 193 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \)
11$C_2 \wr S_4$ \( 1 - 67 T + 5266 T^{2} - 195651 T^{3} + 9393674 T^{4} - 195651 p^{3} T^{5} + 5266 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8} \)
17$C_2 \wr S_4$ \( 1 - 65 T + 40 p^{2} T^{2} - 208599 T^{3} + 50373854 T^{4} - 208599 p^{3} T^{5} + 40 p^{8} T^{6} - 65 p^{9} T^{7} + p^{12} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 24 T + 19587 T^{2} - 398416 T^{3} + 181103520 T^{4} - 398416 p^{3} T^{5} + 19587 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \)
23$C_2 \wr S_4$ \( 1 - 24 T + 4827 T^{2} - 593376 T^{3} - 25012088 T^{4} - 593376 p^{3} T^{5} + 4827 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \)
29$C_2 \wr S_4$ \( 1 - 106 T + 44545 T^{2} - 4299546 T^{3} + 1558837340 T^{4} - 4299546 p^{3} T^{5} + 44545 p^{6} T^{6} - 106 p^{9} T^{7} + p^{12} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 35 T + 92432 T^{2} + 2854743 T^{3} + 3883887326 T^{4} + 2854743 p^{3} T^{5} + 92432 p^{6} T^{6} + 35 p^{9} T^{7} + p^{12} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 97 T + 196464 T^{2} - 13875643 T^{3} + 14750113598 T^{4} - 13875643 p^{3} T^{5} + 196464 p^{6} T^{6} - 97 p^{9} T^{7} + p^{12} T^{8} \)
41$C_2 \wr S_4$ \( 1 - 390 T + 169392 T^{2} - 59939994 T^{3} + 17545316446 T^{4} - 59939994 p^{3} T^{5} + 169392 p^{6} T^{6} - 390 p^{9} T^{7} + p^{12} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 498 T + 65717 T^{2} + 4397718 T^{3} - 980860644 T^{4} + 4397718 p^{3} T^{5} + 65717 p^{6} T^{6} - 498 p^{9} T^{7} + p^{12} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 283 T + 162636 T^{2} - 13616055 T^{3} + 9694562278 T^{4} - 13616055 p^{3} T^{5} + 162636 p^{6} T^{6} - 283 p^{9} T^{7} + p^{12} T^{8} \)
53$C_2 \wr S_4$ \( 1 - 49 T + 3666 p T^{2} - 79623027 T^{3} + 13168244074 T^{4} - 79623027 p^{3} T^{5} + 3666 p^{7} T^{6} - 49 p^{9} T^{7} + p^{12} T^{8} \)
59$C_2 \wr S_4$ \( 1 - 508 T + 300388 T^{2} - 108256572 T^{3} + 79447861574 T^{4} - 108256572 p^{3} T^{5} + 300388 p^{6} T^{6} - 508 p^{9} T^{7} + p^{12} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 611 T + 411342 T^{2} + 218262737 T^{3} + 67578033602 T^{4} + 218262737 p^{3} T^{5} + 411342 p^{6} T^{6} + 611 p^{9} T^{7} + p^{12} T^{8} \)
67$C_2 \wr S_4$ \( 1 - 344 T + 719480 T^{2} - 392638200 T^{3} + 248054931550 T^{4} - 392638200 p^{3} T^{5} + 719480 p^{6} T^{6} - 344 p^{9} T^{7} + p^{12} T^{8} \)
71$C_2 \wr S_4$ \( 1 - 412 T + 138708 T^{2} + 127787028 T^{3} + 2979526390 T^{4} + 127787028 p^{3} T^{5} + 138708 p^{6} T^{6} - 412 p^{9} T^{7} + p^{12} T^{8} \)
73$C_2 \wr S_4$ \( 1 + 338 T + 1554345 T^{2} + 5322506 p T^{3} + 906473517764 T^{4} + 5322506 p^{4} T^{5} + 1554345 p^{6} T^{6} + 338 p^{9} T^{7} + p^{12} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 533 T + 791118 T^{2} + 183770417 T^{3} + 229926206114 T^{4} + 183770417 p^{3} T^{5} + 791118 p^{6} T^{6} + 533 p^{9} T^{7} + p^{12} T^{8} \)
83$C_2 \wr S_4$ \( 1 + 283 T + 1231438 T^{2} + 307666515 T^{3} + 1003775471666 T^{4} + 307666515 p^{3} T^{5} + 1231438 p^{6} T^{6} + 283 p^{9} T^{7} + p^{12} T^{8} \)
89$C_2 \wr S_4$ \( 1 + 629 T + 2648950 T^{2} + 1211614539 T^{3} + 2739597907970 T^{4} + 1211614539 p^{3} T^{5} + 2648950 p^{6} T^{6} + 629 p^{9} T^{7} + p^{12} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 501 T + 3087690 T^{2} + 1496999819 T^{3} + 3961899189546 T^{4} + 1496999819 p^{3} T^{5} + 3087690 p^{6} T^{6} + 501 p^{9} T^{7} + p^{12} T^{8} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.49911965404543354648502227437, −6.93357936204774067299777611303, −6.73737349372174431165645018632, −6.71799133817720698014361867884, −6.37779092644844826724484812654, −5.90239339950019598091450386850, −5.73585873643278905628316881748, −5.67592272470935416229934167147, −5.55043556326618171624325945604, −4.83358522283012306808078147742, −4.74590883977417248278892288336, −4.58080287783492193337600005361, −4.15841288382082115862289636866, −3.99589144083951339182520703898, −3.71016219246673433024269128122, −3.61339628529864423173699172281, −3.48914441480496925478459242893, −2.70720640251054742571471533864, −2.61194967058109358307618663247, −2.45816906165170283870221694042, −2.25835569127809853488644506579, −1.57597336070900194099206701286, −1.34067712607109347442410613515, −1.27094521264281078490520139368, −1.03216265019870539407142462381, 1.03216265019870539407142462381, 1.27094521264281078490520139368, 1.34067712607109347442410613515, 1.57597336070900194099206701286, 2.25835569127809853488644506579, 2.45816906165170283870221694042, 2.61194967058109358307618663247, 2.70720640251054742571471533864, 3.48914441480496925478459242893, 3.61339628529864423173699172281, 3.71016219246673433024269128122, 3.99589144083951339182520703898, 4.15841288382082115862289636866, 4.58080287783492193337600005361, 4.74590883977417248278892288336, 4.83358522283012306808078147742, 5.55043556326618171624325945604, 5.67592272470935416229934167147, 5.73585873643278905628316881748, 5.90239339950019598091450386850, 6.37779092644844826724484812654, 6.71799133817720698014361867884, 6.73737349372174431165645018632, 6.93357936204774067299777611303, 7.49911965404543354648502227437

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