Properties

Label 2-912-1.1-c5-0-32
Degree $2$
Conductor $912$
Sign $1$
Analytic cond. $146.270$
Root an. cond. $12.0942$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·3-s − 80.1·5-s + 221.·7-s + 81·9-s − 6.83·11-s + 1.03e3·13-s − 721.·15-s + 89.8·17-s − 361·19-s + 1.98e3·21-s − 1.93e3·23-s + 3.29e3·25-s + 729·27-s + 8.95e3·29-s + 152.·31-s − 61.5·33-s − 1.77e4·35-s − 2.97e3·37-s + 9.33e3·39-s − 1.38e4·41-s + 1.29e4·43-s − 6.49e3·45-s + 2.35e4·47-s + 3.20e4·49-s + 808.·51-s + 5.83e3·53-s + 547.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.43·5-s + 1.70·7-s + 0.333·9-s − 0.0170·11-s + 1.70·13-s − 0.827·15-s + 0.0753·17-s − 0.229·19-s + 0.984·21-s − 0.764·23-s + 1.05·25-s + 0.192·27-s + 1.97·29-s + 0.0285·31-s − 0.00983·33-s − 2.44·35-s − 0.356·37-s + 0.982·39-s − 1.28·41-s + 1.06·43-s − 0.477·45-s + 1.55·47-s + 1.90·49-s + 0.0435·51-s + 0.285·53-s + 0.0244·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(146.270\)
Root analytic conductor: \(12.0942\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.081429243\)
\(L(\frac12)\) \(\approx\) \(3.081429243\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 9T \)
19 \( 1 + 361T \)
good5 \( 1 + 80.1T + 3.12e3T^{2} \)
7 \( 1 - 221.T + 1.68e4T^{2} \)
11 \( 1 + 6.83T + 1.61e5T^{2} \)
13 \( 1 - 1.03e3T + 3.71e5T^{2} \)
17 \( 1 - 89.8T + 1.41e6T^{2} \)
23 \( 1 + 1.93e3T + 6.43e6T^{2} \)
29 \( 1 - 8.95e3T + 2.05e7T^{2} \)
31 \( 1 - 152.T + 2.86e7T^{2} \)
37 \( 1 + 2.97e3T + 6.93e7T^{2} \)
41 \( 1 + 1.38e4T + 1.15e8T^{2} \)
43 \( 1 - 1.29e4T + 1.47e8T^{2} \)
47 \( 1 - 2.35e4T + 2.29e8T^{2} \)
53 \( 1 - 5.83e3T + 4.18e8T^{2} \)
59 \( 1 - 5.03e3T + 7.14e8T^{2} \)
61 \( 1 + 5.63e4T + 8.44e8T^{2} \)
67 \( 1 + 4.83e4T + 1.35e9T^{2} \)
71 \( 1 + 7.35e4T + 1.80e9T^{2} \)
73 \( 1 - 5.98e4T + 2.07e9T^{2} \)
79 \( 1 - 3.40e4T + 3.07e9T^{2} \)
83 \( 1 + 7.62e4T + 3.93e9T^{2} \)
89 \( 1 + 1.83e4T + 5.58e9T^{2} \)
97 \( 1 - 5.51e4T + 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.897784268129144951316550507623, −8.353627742909534955952654615350, −7.962552581801861663621041012001, −7.10247662395789378208088052449, −5.87325470423380398872838337761, −4.55983362871247484343840447147, −4.12567291737108379475055083564, −3.10198549590267424408913053670, −1.72004815440458286339452448643, −0.811007455588589758684490061343, 0.811007455588589758684490061343, 1.72004815440458286339452448643, 3.10198549590267424408913053670, 4.12567291737108379475055083564, 4.55983362871247484343840447147, 5.87325470423380398872838337761, 7.10247662395789378208088052449, 7.962552581801861663621041012001, 8.353627742909534955952654615350, 8.897784268129144951316550507623

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