Properties

Label 2-912-57.56-c1-0-36
Degree $2$
Conductor $912$
Sign $-0.943 + 0.332i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.943 + 0.332i$
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: \((2,\ 912,\ (\ :1/2),\ -0.943 + 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.217721 - 1.27394i\)
\(L(\frac12)\) \(\approx\) \(0.217721 - 1.27394i\)
\(L(\frac{3}{2})\) 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

−9.358581819353955272861639438178, −8.844966891534763731447996480246, −8.053999243532492629827665975352, −7.38086062968160618937879203311, −6.06644051692345301172136881689, −5.68984777965860849972952620676, −4.19514958909161710162985748078, −3.17379438390478724855635606705, −1.85105490044898194042503456436, −0.56269559371047578467566630843, 2.27218448789381555023435913855, 3.10501960608518363710646247425, 4.17101021529907217688289906249, 4.99525727939636799975678945713, 6.29168550720794500664621359888, 7.03478316735657042958477977067, 7.946265843822307648787595259390, 9.091253803798652752921299459683, 9.616180599234108856807964177736, 10.39813385594226099478384383132

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