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Properties

Label 2-912-57.56-c1-0-33
Degree $2$
Conductor $912$
Sign $-0.697 + 0.716i$
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  + (0.171 + 1.72i)3-s − 3.81i·5-s − 2.25·7-s + (−2.94 + 0.589i)9-s + 2.65i·11-s + 2.28i·13-s + (6.57 − 0.652i)15-s − 2.83i·17-s + (−2.80 − 3.33i)19-s + (−0.385 − 3.87i)21-s − 4.55i·23-s − 9.56·25-s + (−1.51 − 4.96i)27-s − 5.88·29-s + 2.46i·31-s + ⋯
L(s)  = 1  + (0.0987 + 0.995i)3-s − 1.70i·5-s − 0.850·7-s + (−0.980 + 0.196i)9-s + 0.800i·11-s + 0.634i·13-s + (1.69 − 0.168i)15-s − 0.687i·17-s + (−0.643 − 0.765i)19-s + (−0.0840 − 0.846i)21-s − 0.949i·23-s − 1.91·25-s + (−0.292 − 0.956i)27-s − 1.09·29-s + 0.442i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.697 + 0.716i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.170841 - 0.404905i\)
\(L(\frac12)\) \(\approx\) \(0.170841 - 0.404905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.171 - 1.72i)T \)
19 \( 1 + (2.80 + 3.33i)T \)
good5 \( 1 + 3.81iT - 5T^{2} \)
7 \( 1 + 2.25T + 7T^{2} \)
11 \( 1 - 2.65iT - 11T^{2} \)
13 \( 1 - 2.28iT - 13T^{2} \)
17 \( 1 + 2.83iT - 17T^{2} \)
23 \( 1 + 4.55iT - 23T^{2} \)
29 \( 1 + 5.88T + 29T^{2} \)
31 \( 1 - 2.46iT - 31T^{2} \)
37 \( 1 + 8.73iT - 37T^{2} \)
41 \( 1 + 5.54T + 41T^{2} \)
43 \( 1 + 11.2T + 43T^{2} \)
47 \( 1 - 0.494iT - 47T^{2} \)
53 \( 1 - 7.27T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 7.44T + 61T^{2} \)
67 \( 1 - 4.50iT - 67T^{2} \)
71 \( 1 - 2.61T + 71T^{2} \)
73 \( 1 + 9.17T + 73T^{2} \)
79 \( 1 + 5.69iT - 79T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 18.9iT - 97T^{2} \)
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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

−9.601193984663693644243599125063, −9.023434175694636076745643504849, −8.558106647672828668362248557186, −7.27434877869960730861245880487, −6.14437916663608931761490887353, −5.01019709386895154409142009482, −4.59862959881982016113570515662, −3.60578163017445106879068104035, −2.12285265911615744198150067071, −0.19022532742368211677500864059, 1.89367377645874448213238401996, 3.17444952280101372096551075926, 3.48523144880268599955669948334, 5.68463365212304917443963161645, 6.27322978934746714468217884399, 6.88842858154683787356062486943, 7.77105687318202608800457814868, 8.470404162085424412964663663141, 9.761437531229896873118505443942, 10.43567804693291889578135881253

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