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Properties

Label 2-912-12.11-c1-0-31
Degree $2$
Conductor $912$
Sign $-0.965 + 0.258i$
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  + (−0.448 − 1.67i)3-s − 2.39i·5-s − 3.59i·7-s + (−2.59 + 1.50i)9-s + 5.96·11-s − 3.34·13-s + (−4.01 + 1.07i)15-s − 6.52i·17-s − i·19-s + (−6.01 + 1.61i)21-s + 3.33·23-s − 0.748·25-s + (3.67 + 3.67i)27-s + 5.66i·29-s + 1.34i·31-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s − 1.07i·5-s − 1.35i·7-s + (−0.865 + 0.500i)9-s + 1.79·11-s − 0.928·13-s + (−1.03 + 0.277i)15-s − 1.58i·17-s − 0.229i·19-s + (−1.31 + 0.352i)21-s + 0.695·23-s − 0.149·25-s + (0.707 + 0.706i)27-s + 1.05i·29-s + 0.241i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.965 + 0.258i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171732 - 1.30386i\)
\(L(\frac12)\) \(\approx\) \(0.171732 - 1.30386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.448 + 1.67i)T \)
19 \( 1 + iT \)
good5 \( 1 + 2.39iT - 5T^{2} \)
7 \( 1 + 3.59iT - 7T^{2} \)
11 \( 1 - 5.96T + 11T^{2} \)
13 \( 1 + 3.34T + 13T^{2} \)
17 \( 1 + 6.52iT - 17T^{2} \)
23 \( 1 - 3.33T + 23T^{2} \)
29 \( 1 - 5.66iT - 29T^{2} \)
31 \( 1 - 1.34iT - 31T^{2} \)
37 \( 1 + 6.84T + 37T^{2} \)
41 \( 1 - 7.68iT - 41T^{2} \)
43 \( 1 + 0.402iT - 43T^{2} \)
47 \( 1 + 1.83T + 47T^{2} \)
53 \( 1 - 1.76iT - 53T^{2} \)
59 \( 1 + 6.22T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 0.300iT - 67T^{2} \)
71 \( 1 - 7.35T + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 2.80iT - 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 + 7.04T + 97T^{2} \)
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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

−9.424676587050273649706267498927, −8.999598324064227931455749814182, −7.84724373486675181336631565362, −7.02101490339637167958296425657, −6.63673607092751161544621307312, −5.13010121972548139558211246457, −4.57341539991812918215685736583, −3.21577606637072535754216073191, −1.49434862878286589990728755927, −0.69082781298897600455353045369, 2.11156089248357925983653876303, 3.27605536108128730022956881504, 4.09877753440020425842255684107, 5.30862846674428456902074811078, 6.20335902981582742799074818327, 6.73369945002316877914661035947, 8.158221189216935546265287519408, 9.085523417442178814239595773417, 9.544729960518735442728987305132, 10.53861642015825056543477910753

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