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Properties

Label 2-912-76.31-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.659 - 0.751i$
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  + (0.5 + 0.866i)3-s + (1.33 + 2.31i)5-s + 3.93i·7-s + (−0.499 + 0.866i)9-s + 2.20i·11-s + (−3.60 − 2.07i)13-s + (−1.33 + 2.31i)15-s + (−0.571 − 0.990i)17-s + (4.24 + 0.990i)19-s + (−3.40 + 1.96i)21-s + (−3.19 − 1.84i)23-s + (−1.07 + 1.85i)25-s − 0.999·27-s + (−2.10 − 1.21i)29-s − 3.67·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.597 + 1.03i)5-s + 1.48i·7-s + (−0.166 + 0.288i)9-s + 0.664i·11-s + (−0.998 − 0.576i)13-s + (−0.345 + 0.597i)15-s + (−0.138 − 0.240i)17-s + (0.973 + 0.227i)19-s + (−0.743 + 0.429i)21-s + (−0.665 − 0.384i)23-s + (−0.214 + 0.371i)25-s − 0.192·27-s + (−0.390 − 0.225i)29-s − 0.659·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689669 + 1.52226i\)
\(L(\frac12)\) \(\approx\) \(0.689669 + 1.52226i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4.24 - 0.990i)T \)
good5 \( 1 + (-1.33 - 2.31i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.93iT - 7T^{2} \)
11 \( 1 - 2.20iT - 11T^{2} \)
13 \( 1 + (3.60 + 2.07i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.571 + 0.990i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.19 + 1.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.10 + 1.21i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.67T + 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 + (-8.01 + 4.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.490 + 0.283i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 - 6.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.09 - 2.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.33 - 2.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.41 - 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.73 - 3.00i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.24 - 10.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.35 - 2.34i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.50 - 6.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.98iT - 83T^{2} \)
89 \( 1 + (-12.9 - 7.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.91 + 1.68i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.30339277205901290066677806992, −9.489443761777844723610556097530, −9.071656991067163940581723497443, −7.79009383448897722475101579011, −7.09635388657609815403737159302, −5.79101644823139780649050409252, −5.43438810842776988111038249973, −4.05835543853070795246952998353, −2.60473091341693368255464569556, −2.38796607972640679783144273192, 0.75830768734567576895811671950, 1.83710953731770869445486813751, 3.37403017923393060480696825084, 4.45619374130239820141823887542, 5.34603602930912444559983602397, 6.43868912507211643522696955499, 7.37306223332144303212504392579, 7.968554667125915064011595229266, 9.089560692833360825630563139507, 9.609863008622265824930978592202

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