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Properties

Label 2-912-912.341-c1-0-90
Degree $2$
Conductor $912$
Sign $0.777 + 0.628i$
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  + (−1.36 − 0.370i)2-s + (0.456 + 1.67i)3-s + (1.72 + 1.01i)4-s + (−2.14 + 2.14i)5-s + (−0.00361 − 2.44i)6-s − 1.36i·7-s + (−1.97 − 2.02i)8-s + (−2.58 + 1.52i)9-s + (3.72 − 2.13i)10-s + (4.29 − 4.29i)11-s + (−0.902 + 3.34i)12-s + (−1.95 − 1.95i)13-s + (−0.506 + 1.86i)14-s + (−4.57 − 2.61i)15-s + (1.95 + 3.49i)16-s − 4.16i·17-s + ⋯
L(s)  = 1  + (−0.965 − 0.262i)2-s + (0.263 + 0.964i)3-s + (0.862 + 0.505i)4-s + (−0.961 + 0.961i)5-s + (−0.00147 − 0.999i)6-s − 0.516i·7-s + (−0.699 − 0.714i)8-s + (−0.861 + 0.508i)9-s + (1.17 − 0.675i)10-s + (1.29 − 1.29i)11-s + (−0.260 + 0.965i)12-s + (−0.542 − 0.542i)13-s + (−0.135 + 0.498i)14-s + (−1.18 − 0.673i)15-s + (0.488 + 0.872i)16-s − 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.777 + 0.628i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.777 + 0.628i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670673 - 0.237165i\)
\(L(\frac12)\) \(\approx\) \(0.670673 - 0.237165i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.370i)T \)
3 \( 1 + (-0.456 - 1.67i)T \)
19 \( 1 + (2.11 + 3.81i)T \)
good5 \( 1 + (2.14 - 2.14i)T - 5iT^{2} \)
7 \( 1 + 1.36iT - 7T^{2} \)
11 \( 1 + (-4.29 + 4.29i)T - 11iT^{2} \)
13 \( 1 + (1.95 + 1.95i)T + 13iT^{2} \)
17 \( 1 + 4.16iT - 17T^{2} \)
23 \( 1 - 6.21T + 23T^{2} \)
29 \( 1 + (4.85 - 4.85i)T - 29iT^{2} \)
31 \( 1 + 5.08iT - 31T^{2} \)
37 \( 1 + (-2.68 + 2.68i)T - 37iT^{2} \)
41 \( 1 + 3.35iT - 41T^{2} \)
43 \( 1 + (4.09 + 4.09i)T + 43iT^{2} \)
47 \( 1 - 1.02iT - 47T^{2} \)
53 \( 1 + (-2.02 - 2.02i)T + 53iT^{2} \)
59 \( 1 + (-10.4 - 10.4i)T + 59iT^{2} \)
61 \( 1 + (7.60 - 7.60i)T - 61iT^{2} \)
67 \( 1 + (-2.86 - 2.86i)T + 67iT^{2} \)
71 \( 1 - 1.11iT - 71T^{2} \)
73 \( 1 + 1.67iT - 73T^{2} \)
79 \( 1 - 2.48iT - 79T^{2} \)
83 \( 1 + (11.5 + 11.5i)T + 83iT^{2} \)
89 \( 1 + 17.2iT - 89T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−10.08012278724120905844610570738, −9.013664755055797462545642537791, −8.710254257377896718859287328988, −7.38171294433293853343880117489, −7.09220629766850934599783466750, −5.77275967466489449624312521568, −4.23744828661411349223591417207, −3.38751305813747448769389968235, −2.77238852745252779024814895712, −0.49625212628073683816763304498, 1.25370446682175900300167243886, 2.11778935130053300233143673047, 3.79463769644039347924311047258, 5.01896520395390600328246356125, 6.31528792640586416846498519803, 6.93721636105465450052540501218, 7.83493583866830347858276434649, 8.433303124332266516599486427953, 9.133918764864428773593344621540, 9.772140776984544767110486536372

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