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Properties

Label 2-912-57.56-c1-0-8
Degree $2$
Conductor $912$
Sign $0.606 - 0.794i$
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  + (−1.58 − 0.707i)3-s + 2.23i·5-s + 7-s + (2.00 + 2.23i)9-s − 2.23i·11-s − 4.24i·13-s + (1.58 − 3.53i)15-s + 2.23i·17-s + (−1 + 4.24i)19-s + (−1.58 − 0.707i)21-s + 4.47i·23-s + (−1.58 − 4.94i)27-s + 4.24i·31-s + (−1.58 + 3.53i)33-s + 2.23i·35-s + ⋯
L(s)  = 1  + (−0.912 − 0.408i)3-s + 0.999i·5-s + 0.377·7-s + (0.666 + 0.745i)9-s − 0.674i·11-s − 1.17i·13-s + (0.408 − 0.912i)15-s + 0.542i·17-s + (−0.229 + 0.973i)19-s + (−0.345 − 0.154i)21-s + 0.932i·23-s + (−0.304 − 0.952i)27-s + 0.762i·31-s + (−0.275 + 0.615i)33-s + 0.377i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.606 - 0.794i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984512 + 0.487030i\)
\(L(\frac12)\) \(\approx\) \(0.984512 + 0.487030i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.707i)T \)
19 \( 1 + (1 - 4.24i)T \)
good5 \( 1 - 2.23iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 2.23iT - 17T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 - 9.48T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 - 9.48T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−10.59092633645048983370819682278, −9.632921044155280373860561905522, −8.169317183505829865204911218770, −7.71194210846944611068627894942, −6.66102578868293613389154649559, −5.94450039628884566817835354187, −5.22611103466608517846201326038, −3.86482989066362919673562776764, −2.73280541175669526862216654760, −1.23007196348289052772988160560, 0.68363830704138148030605529812, 2.15996640133711244822286571388, 4.11601989922552432960035114644, 4.65181933028358657893359928436, 5.37321178214208392813096354572, 6.52740168134791730591883487123, 7.24037430816403104534538481196, 8.487363595749384958201571980732, 9.298039000950643963906455288054, 9.807213296822868958228757255406

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