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Properties

Label 2-912-57.56-c1-0-12
Degree $2$
Conductor $912$
Sign $0.596 - 0.802i$
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  + (1.5 − 0.866i)3-s + 3.46i·5-s − 7-s + (1.5 − 2.59i)9-s + 3.46i·11-s − 1.73i·13-s + (2.99 + 5.19i)15-s + 1.73i·17-s + (4 − 1.73i)19-s + (−1.5 + 0.866i)21-s + 5.19i·23-s − 6.99·25-s − 5.19i·27-s + 9·29-s + 10.3i·31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + 1.54i·5-s − 0.377·7-s + (0.5 − 0.866i)9-s + 1.04i·11-s − 0.480i·13-s + (0.774 + 1.34i)15-s + 0.420i·17-s + (0.917 − 0.397i)19-s + (−0.327 + 0.188i)21-s + 1.08i·23-s − 1.39·25-s − 0.999i·27-s + 1.67·29-s + 1.86i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.596 - 0.802i$
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: \((2,\ 912,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79987 + 0.905505i\)
\(L(\frac12)\) \(\approx\) \(1.79987 + 0.905505i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 - 10.3iT - 31T^{2} \)
37 \( 1 - 6.92iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 9T + 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−10.03442972649494056335382487408, −9.602994139561981452123843556916, −8.386583985166649078361424791657, −7.54907452133010029137486358438, −6.88067588100079673853203211155, −6.35130228305099953158678794439, −4.84615344726309464006343551243, −3.31130025298000745540489865115, −3.02482974477889927597006412686, −1.69028020783213549665740772792, 0.923466411521167865382362748965, 2.48557632430000821629214565507, 3.68828982477715838594221815009, 4.56078980769246016118577449845, 5.36289130826234626919889280487, 6.47517911242109154942842088453, 7.919638778770828429176326313830, 8.302578397461485617962229000044, 9.294512334634324008400509642944, 9.515191572864907516077214823731

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