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Properties

Label 2-912-57.50-c1-0-25
Degree $2$
Conductor $912$
Sign $0.999 + 0.0257i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 + 0.480i)3-s + (2.52 − 1.45i)5-s − 0.106·7-s + (2.53 + 1.59i)9-s + 0.387i·11-s + (−0.750 − 0.433i)13-s + (4.90 − 1.21i)15-s + (2.03 − 1.17i)17-s + (1.50 + 4.09i)19-s + (−0.177 − 0.0512i)21-s + (−4.19 − 2.41i)23-s + (1.75 − 3.04i)25-s + (3.45 + 3.87i)27-s + (4.23 − 7.32i)29-s + 7.85i·31-s + ⋯
L(s)  = 1  + (0.960 + 0.277i)3-s + (1.13 − 0.652i)5-s − 0.0403·7-s + (0.846 + 0.532i)9-s + 0.116i·11-s + (−0.208 − 0.120i)13-s + (1.26 − 0.313i)15-s + (0.493 − 0.285i)17-s + (0.345 + 0.938i)19-s + (−0.0387 − 0.0111i)21-s + (−0.873 − 0.504i)23-s + (0.351 − 0.609i)25-s + (0.665 + 0.746i)27-s + (0.785 − 1.36i)29-s + 1.41i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.999 + 0.0257i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.999 + 0.0257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68511 - 0.0346356i\)
\(L(\frac12)\) \(\approx\) \(2.68511 - 0.0346356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.66 - 0.480i)T \)
19 \( 1 + (-1.50 - 4.09i)T \)
good5 \( 1 + (-2.52 + 1.45i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.106T + 7T^{2} \)
11 \( 1 - 0.387iT - 11T^{2} \)
13 \( 1 + (0.750 + 0.433i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.03 + 1.17i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.19 + 2.41i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.23 + 7.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.85iT - 31T^{2} \)
37 \( 1 + 0.670iT - 37T^{2} \)
41 \( 1 + (0.717 + 1.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.67 + 6.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.41 + 1.97i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.76 + 6.52i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.53 - 6.11i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.65 - 9.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.42 + 3.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.47 + 6.01i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.57 - 6.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.2 + 7.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.15iT - 83T^{2} \)
89 \( 1 + (8.09 - 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.56 - 2.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04629363701400514615402698852, −9.316254544769405856061714041745, −8.490344002861438640101035379443, −7.81856424597493508418067570402, −6.67081814639806278054290327932, −5.60927951677404130255058576751, −4.79739829361303468355229576914, −3.67620668573289170980714368352, −2.48382881943502298536651807914, −1.48494908513339128786671337427, 1.54922096208578587240228100095, 2.58570500485774716255509399269, 3.40732984115932228306822078459, 4.75373336786289454337724210700, 5.98949502207444959828162261299, 6.68745077499971231594593747815, 7.57956434707481290227922095253, 8.423418393815061237034310564225, 9.497587324322991026903336359904, 9.789811034329135023091004880556

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