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Properties

Label 2-912-19.11-c1-0-11
Degree $2$
Conductor $912$
Sign $-0.0547 + 0.998i$
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 + (−0.347 + 0.601i)5-s − 0.305·7-s + (−0.499 − 0.866i)9-s − 4.82·11-s + (−0.5 − 0.866i)13-s + (−0.347 − 0.601i)15-s + (3.75 − 6.51i)17-s + (−3.06 − 3.10i)19-s + (0.152 − 0.264i)21-s + (0.347 + 0.601i)23-s + (2.25 + 3.91i)25-s + 0.999·27-s + (−5.06 − 8.77i)29-s + 1.82·31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.155 + 0.269i)5-s − 0.115·7-s + (−0.166 − 0.288i)9-s − 1.45·11-s + (−0.138 − 0.240i)13-s + (−0.0896 − 0.155i)15-s + (0.911 − 1.57i)17-s + (−0.702 − 0.711i)19-s + (0.0333 − 0.0577i)21-s + (0.0724 + 0.125i)23-s + (0.451 + 0.782i)25-s + 0.192·27-s + (−0.940 − 1.62i)29-s + 0.327·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0547 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.0547 + 0.998i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.0547 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.439099 - 0.463836i\)
\(L(\frac12)\) \(\approx\) \(0.439099 - 0.463836i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.06 + 3.10i)T \)
good5 \( 1 + (0.347 - 0.601i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.305T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.75 + 6.51i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.347 - 0.601i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.06 + 8.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 - 6.51T + 37T^{2} \)
41 \( 1 + (2.69 - 4.66i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.84 + 3.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.71 + 4.70i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.04 - 3.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.194 + 0.337i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.91 + 6.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.45 + 9.44i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.19 + 3.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.21 + 12.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.739T + 83T^{2} \)
89 \( 1 + (-0.411 - 0.712i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.45 - 9.44i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.848509300436939154452492803711, −9.316216249191500237454264158057, −8.022549438326216688213725493691, −7.47598235686626849333656655651, −6.36882267425442693014108031361, −5.31165621003786517263243180217, −4.74141306071808377687636105008, −3.36009794106497061505609984199, −2.51128029913974098750449636693, −0.31115395571488640387238371104, 1.49698808146477888443887941193, 2.79326782988793334970850667669, 4.06613190895187954541176222906, 5.20283934819457970387306585801, 5.94932477960050753670725821251, 6.88814600394127633094564729387, 8.060460914369064601028641455264, 8.212550068170988880274787976638, 9.540841738120835273281296352547, 10.53935208926570318495688585220

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