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Properties

Label 2-912-76.67-c1-0-9
Degree $2$
Conductor $912$
Sign $0.808 + 0.588i$
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  + (0.766 − 0.642i)3-s + (−0.800 − 0.291i)5-s + (1.70 + 0.983i)7-s + (0.173 − 0.984i)9-s + (0.742 − 0.428i)11-s + (2.79 − 3.33i)13-s + (−0.800 + 0.291i)15-s + (1.26 + 7.14i)17-s + (−0.199 − 4.35i)19-s + (1.93 − 0.341i)21-s + (−0.620 − 1.70i)23-s + (−3.27 − 2.74i)25-s + (−0.500 − 0.866i)27-s + (9.99 + 1.76i)29-s + (1.88 − 3.25i)31-s + ⋯
L(s)  = 1  + (0.442 − 0.371i)3-s + (−0.357 − 0.130i)5-s + (0.644 + 0.371i)7-s + (0.0578 − 0.328i)9-s + (0.223 − 0.129i)11-s + (0.776 − 0.925i)13-s + (−0.206 + 0.0752i)15-s + (0.305 + 1.73i)17-s + (−0.0456 − 0.998i)19-s + (0.422 − 0.0745i)21-s + (−0.129 − 0.355i)23-s + (−0.654 − 0.549i)25-s + (−0.0962 − 0.166i)27-s + (1.85 + 0.327i)29-s + (0.337 − 0.584i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.808 + 0.588i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.808 + 0.588i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84001 - 0.598804i\)
\(L(\frac12)\) \(\approx\) \(1.84001 - 0.598804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.199 + 4.35i)T \)
good5 \( 1 + (0.800 + 0.291i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-1.70 - 0.983i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.742 + 0.428i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.79 + 3.33i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.26 - 7.14i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (0.620 + 1.70i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-9.99 - 1.76i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-1.88 + 3.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.42iT - 37T^{2} \)
41 \( 1 + (0.985 + 1.17i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.98 + 10.9i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-7.46 - 1.31i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.86 - 5.11i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.02 + 5.83i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.62 + 0.955i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.41 - 8.04i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (11.8 + 4.30i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.45 + 3.73i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (-3.58 + 3.01i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (11.1 + 6.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.05 - 7.22i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.66 - 1.35i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.15973322632548261582599506930, −8.715151738959087959328822863907, −8.454323163386452906752639818796, −7.72811487540328012261815623941, −6.53768681843617579619510823973, −5.78487160658492569281844708639, −4.57542030125489299537111098606, −3.60130199955849802294379633800, −2.42799284702792288113610477589, −1.07469825963907722083178554511, 1.36390928864109613270305623063, 2.81939295148710106032562684281, 3.96339414952290184043434035474, 4.61823942427081217217357921308, 5.78472456942684470863770999560, 6.96424559668207579427637905977, 7.69794632152009610723606735751, 8.494727685909068399689719344275, 9.368129779568576217295638784504, 10.06882935067672149836938340129

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