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Properties

Label 2-912-304.125-c1-0-10
Degree $2$
Conductor $912$
Sign $-0.835 + 0.549i$
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  + (0.682 + 1.23i)2-s + (−0.965 + 0.258i)3-s + (−1.06 + 1.69i)4-s + (0.477 + 1.78i)5-s + (−0.979 − 1.02i)6-s + 2.71i·7-s + (−2.82 − 0.172i)8-s + (0.866 − 0.499i)9-s + (−1.88 + 1.80i)10-s + (−0.0277 + 0.0277i)11-s + (0.595 − 1.90i)12-s + (−0.894 + 3.33i)13-s + (−3.36 + 1.85i)14-s + (−0.922 − 1.59i)15-s + (−1.71 − 3.61i)16-s + (−1.50 + 2.61i)17-s + ⋯
L(s)  = 1  + (0.482 + 0.875i)2-s + (−0.557 + 0.149i)3-s + (−0.534 + 0.845i)4-s + (0.213 + 0.796i)5-s + (−0.399 − 0.416i)6-s + 1.02i·7-s + (−0.998 − 0.0608i)8-s + (0.288 − 0.166i)9-s + (−0.595 + 0.571i)10-s + (−0.00836 + 0.00836i)11-s + (0.171 − 0.551i)12-s + (−0.248 + 0.925i)13-s + (−0.898 + 0.494i)14-s + (−0.238 − 0.412i)15-s + (−0.428 − 0.903i)16-s + (−0.365 + 0.633i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.835 + 0.549i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342113 - 1.14327i\)
\(L(\frac12)\) \(\approx\) \(0.342113 - 1.14327i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.682 - 1.23i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (-2.90 + 3.25i)T \)
good5 \( 1 + (-0.477 - 1.78i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 - 2.71iT - 7T^{2} \)
11 \( 1 + (0.0277 - 0.0277i)T - 11iT^{2} \)
13 \( 1 + (0.894 - 3.33i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.50 - 2.61i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.0133 + 0.00771i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.740 + 2.76i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + 5.66T + 31T^{2} \)
37 \( 1 + (6.26 - 6.26i)T - 37iT^{2} \)
41 \( 1 + (-6.55 - 3.78i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.588 + 2.19i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.16 + 5.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.05 + 2.15i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.78 + 6.65i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.50 - 9.34i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.35 + 5.04i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.84 + 2.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12.3 - 7.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.99 + 3.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.04 - 4.04i)T + 83iT^{2} \)
89 \( 1 + (-14.6 + 8.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.92 - 11.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.66652318391720760915567440288, −9.516668396000985925821578543372, −8.946364831536160560027396668658, −7.890417874362649939938863781110, −6.79784545316197778012019176396, −6.41759463824933024022836423157, −5.44278710146978278845612083670, −4.67874652387391248284811219726, −3.47328330946747472091080672166, −2.31701036712614552310197434381, 0.52692908268278994295250054898, 1.58876947472786205886767953797, 3.16784432598694405651315606808, 4.22944294962344610674624161031, 5.12783172768967343311662890666, 5.70875870925163321135486085250, 6.95748618580875385526567203299, 7.88565575393531745409657731841, 9.114089104253188315794166680888, 9.744537338254690844612265446954

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