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Properties

Label 2-912-57.56-c1-0-5
Degree $2$
Conductor $912$
Sign $-0.381 - 0.924i$
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  + (−0.707 + 1.58i)3-s − 2.23i·5-s − 7-s + (−2.00 − 2.23i)9-s − 2.23i·11-s + 3.16i·13-s + (3.53 + 1.58i)15-s + 6.70i·17-s + (−3 + 3.16i)19-s + (0.707 − 1.58i)21-s + 4.47i·23-s + (4.94 − 1.58i)27-s + 5.65·29-s + 3.16i·31-s + (3.53 + 1.58i)33-s + ⋯
L(s)  = 1  + (−0.408 + 0.912i)3-s − 0.999i·5-s − 0.377·7-s + (−0.666 − 0.745i)9-s − 0.674i·11-s + 0.877i·13-s + (0.912 + 0.408i)15-s + 1.62i·17-s + (−0.688 + 0.725i)19-s + (0.154 − 0.345i)21-s + 0.932i·23-s + (0.952 − 0.304i)27-s + 1.05·29-s + 0.567i·31-s + (0.615 + 0.275i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.381 - 0.924i$
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: \((2,\ 912,\ (\ :1/2),\ -0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482394 + 0.720778i\)
\(L(\frac12)\) \(\approx\) \(0.482394 + 0.720778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 1.58i)T \)
19 \( 1 + (3 - 3.16i)T \)
good5 \( 1 + 2.23iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 + 2.23iT - 11T^{2} \)
13 \( 1 - 3.16iT - 13T^{2} \)
17 \( 1 - 6.70iT - 17T^{2} \)
23 \( 1 - 4.47iT - 23T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 - 3.16iT - 31T^{2} \)
37 \( 1 - 9.48iT - 37T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 6.32iT - 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 - 12.6iT - 79T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 3.16iT - 97T^{2} \)
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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

−10.28075874266891010725600487646, −9.587837356339651592144247287874, −8.598598880898341357460979413473, −8.338857085958036522248221365306, −6.61383618787241119710815369609, −5.97421087106223824009225680366, −4.99759909603623267542343448495, −4.17062025127237771486620201882, −3.30853017385887234020676798615, −1.42286338387552004725334514200, 0.44714780020650638930014820179, 2.35457229626279657845647034775, 2.98842393226924914420396648086, 4.62411563386382750183457771526, 5.62277476298258352768183391930, 6.74268050916995412554314943872, 6.98132831895399362915872014492, 7.916194677846779579613007855675, 8.937099936863126864208714465678, 10.05128273705276275378165049658

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