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Properties

Label 2-912-57.8-c1-0-31
Degree $2$
Conductor $912$
Sign $0.484 + 0.874i$
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  + (1.18 − 1.26i)3-s + (0.686 + 0.396i)5-s + 2.37·7-s + (−0.186 − 2.99i)9-s − 3.46i·11-s + (−2.87 + 1.65i)13-s + (1.31 − 0.396i)15-s + (0.686 + 0.396i)17-s + (4 + 1.73i)19-s + (2.81 − 2.99i)21-s + (6.43 − 3.71i)23-s + (−2.18 − 3.78i)25-s + (−4.00 − 3.31i)27-s + (−2.68 − 4.65i)29-s + 4.40i·31-s + ⋯
L(s)  = 1  + (0.684 − 0.728i)3-s + (0.306 + 0.177i)5-s + 0.896·7-s + (−0.0620 − 0.998i)9-s − 1.04i·11-s + (−0.796 + 0.459i)13-s + (0.339 − 0.102i)15-s + (0.166 + 0.0960i)17-s + (0.917 + 0.397i)19-s + (0.614 − 0.653i)21-s + (1.34 − 0.774i)23-s + (−0.437 − 0.757i)25-s + (−0.769 − 0.638i)27-s + (−0.498 − 0.863i)29-s + 0.790i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.484 + 0.874i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94082 - 1.14381i\)
\(L(\frac12)\) \(\approx\) \(1.94082 - 1.14381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 + 1.26i)T \)
19 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-0.686 - 0.396i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.37T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (2.87 - 1.65i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.686 - 0.396i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-6.43 + 3.71i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.68 + 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.40iT - 31T^{2} \)
37 \( 1 - 7.86iT - 37T^{2} \)
41 \( 1 + (0.313 - 0.543i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.127 + 0.221i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.68 + 2.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.68 - 9.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.68 + 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.2 - 5.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.31 + 5.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.87 + 4.97i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.98 + 5.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.46iT - 83T^{2} \)
89 \( 1 + (-3.68 - 6.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 6.38i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.832849306492092429710402171332, −8.945965418591055021936772507753, −8.245693704330728005484107264758, −7.51752990345676080537005093523, −6.64496481327307229168511369872, −5.70634139517173963621952683379, −4.60808663376182853960928729684, −3.29560626665196015554460041421, −2.34350109767437656404990814574, −1.11150631273840534052389915597, 1.68142902616280834193807810677, 2.78260059685038938493390978920, 3.97109712721241513517429721697, 5.10026514747757044155259599344, 5.35027901591189973234906641610, 7.39215106665300541929020180618, 7.47473533498303976394752899483, 8.768281478334844893487895876157, 9.439346058757288646143455451382, 10.00680024252806936716576032271

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