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Properties

Label 2-912-57.50-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.934 - 0.354i$
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  + (−0.712 + 1.57i)3-s + (1.02 − 0.594i)5-s − 1.04·7-s + (−1.98 − 2.24i)9-s + 3.41i·11-s + (0.793 + 0.458i)13-s + (0.205 + 2.04i)15-s + (−2.59 + 1.49i)17-s + (−4.30 + 0.688i)19-s + (0.744 − 1.65i)21-s + (−1.62 − 0.940i)23-s + (−1.79 + 3.10i)25-s + (4.96 − 1.53i)27-s + (0.797 − 1.38i)29-s + 8.98i·31-s + ⋯
L(s)  = 1  + (−0.411 + 0.911i)3-s + (0.460 − 0.265i)5-s − 0.395·7-s + (−0.662 − 0.749i)9-s + 1.02i·11-s + (0.220 + 0.127i)13-s + (0.0530 + 0.529i)15-s + (−0.628 + 0.363i)17-s + (−0.987 + 0.157i)19-s + (0.162 − 0.360i)21-s + (−0.339 − 0.196i)23-s + (−0.358 + 0.621i)25-s + (0.955 − 0.295i)27-s + (0.148 − 0.256i)29-s + 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.934 - 0.354i$
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: \((2,\ 912,\ (\ :1/2),\ -0.934 - 0.354i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.131268 + 0.715477i\)
\(L(\frac12)\) \(\approx\) \(0.131268 + 0.715477i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.712 - 1.57i)T \)
19 \( 1 + (4.30 - 0.688i)T \)
good5 \( 1 + (-1.02 + 0.594i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.04T + 7T^{2} \)
11 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-0.793 - 0.458i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.59 - 1.49i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.62 + 0.940i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.797 + 1.38i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.98iT - 31T^{2} \)
37 \( 1 - 4.54iT - 37T^{2} \)
41 \( 1 + (0.469 + 0.813i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.73 - 4.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.82 + 4.51i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0418 + 0.0725i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.766 - 1.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.67 + 5.01i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.80 + 3.13i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.38 - 7.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.95 + 1.70i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.04iT - 83T^{2} \)
89 \( 1 + (2.48 - 4.31i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 - 4.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35176760190306788774612748923, −9.703699340212452644100863293142, −9.024218669637384928289478402906, −8.153372542127940289684435417337, −6.74064762951031877408616196035, −6.18694643575730906966675806715, −5.05126902429586797991533838000, −4.38654097635066787110300594148, −3.29756400814093123837116114442, −1.83351940434828350892784113658, 0.34179327058085414914310807954, 1.98332182366877589022990121446, 2.98275655596944376388081338382, 4.39095466197766168915924001499, 5.80328949497280361682199860561, 6.14411744580016139659878007394, 7.02357907371800589416034780221, 8.017986065918161946891125125905, 8.755970566623143478511028389633, 9.753923036708208647504949955991

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