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Properties

Label 2-912-57.8-c1-0-35
Degree $2$
Conductor $912$
Sign $-0.547 + 0.836i$
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  + (0.956 − 1.44i)3-s + (−1.67 − 0.966i)5-s + 1.30·7-s + (−1.16 − 2.76i)9-s − 0.793i·11-s + (5.90 − 3.41i)13-s + (−2.99 + 1.49i)15-s + (−3.61 − 2.08i)17-s + (−0.402 + 4.34i)19-s + (1.24 − 1.87i)21-s + (−3.32 + 1.92i)23-s + (−0.631 − 1.09i)25-s + (−5.10 − 0.954i)27-s + (−0.449 − 0.778i)29-s − 5.18i·31-s + ⋯
L(s)  = 1  + (0.552 − 0.833i)3-s + (−0.748 − 0.432i)5-s + 0.491·7-s + (−0.389 − 0.920i)9-s − 0.239i·11-s + (1.63 − 0.946i)13-s + (−0.773 + 0.385i)15-s + (−0.875 − 0.505i)17-s + (−0.0923 + 0.995i)19-s + (0.271 − 0.409i)21-s + (−0.693 + 0.400i)23-s + (−0.126 − 0.218i)25-s + (−0.982 − 0.183i)27-s + (−0.0834 − 0.144i)29-s − 0.930i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.547 + 0.836i$
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: \((2,\ 912,\ (\ :1/2),\ -0.547 + 0.836i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.737509 - 1.36458i\)
\(L(\frac12)\) \(\approx\) \(0.737509 - 1.36458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.956 + 1.44i)T \)
19 \( 1 + (0.402 - 4.34i)T \)
good5 \( 1 + (1.67 + 0.966i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 + 0.793iT - 11T^{2} \)
13 \( 1 + (-5.90 + 3.41i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.61 + 2.08i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.32 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.449 + 0.778i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.18iT - 31T^{2} \)
37 \( 1 + 1.51iT - 37T^{2} \)
41 \( 1 + (-4.52 + 7.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.92 - 6.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.48 + 3.74i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.22 + 7.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.84 - 8.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.28 - 2.23i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.5 + 6.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.99 + 8.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.83 - 6.64i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.00 - 0.580i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (-1.30 - 2.26i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.72 + 1.57i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.620058382187807505488961293799, −8.583138903027370714259630877965, −8.173090643866011627223045972653, −7.57198662987426588271171057176, −6.36139157224437748187927559536, −5.62777045760032636104526778464, −4.16234789686364949740062702825, −3.43657855139915421968792023668, −2.01077835997530487292246036985, −0.70333691842555043205251089225, 1.88197825657912727598145549154, 3.23530438632842535916459198979, 4.10936634350289781377390859644, 4.72968026745066401770017625673, 6.12265905212599692972871272210, 7.04865143855702881995568063537, 8.119234183941276936831577425972, 8.675264835218578831557836145263, 9.399260628626983915806588220273, 10.55308079277149287467715228026

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