Properties

Label 2-1148-287.174-c0-0-0
Degree $2$
Conductor $1148$
Sign $0.561 - 0.827i$
Analytic cond. $0.572926$
Root an. cond. $0.756919$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + (0.951 − 0.309i)5-s + (−0.809 + 0.587i)7-s + (0.5 − 1.53i)11-s + (−0.587 + 0.809i)13-s + (0.309 + 0.951i)15-s + (1.53 + 0.5i)17-s + (0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (−0.5 − 0.363i)23-s + i·27-s + (−0.190 − 0.587i)29-s + (−0.587 − 0.190i)31-s + (1.53 + 0.5i)33-s + (−0.587 + 0.809i)35-s + ⋯
L(s)  = 1  + i·3-s + (0.951 − 0.309i)5-s + (−0.809 + 0.587i)7-s + (0.5 − 1.53i)11-s + (−0.587 + 0.809i)13-s + (0.309 + 0.951i)15-s + (1.53 + 0.5i)17-s + (0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (−0.5 − 0.363i)23-s + i·27-s + (−0.190 − 0.587i)29-s + (−0.587 − 0.190i)31-s + (1.53 + 0.5i)33-s + (−0.587 + 0.809i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 - 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.561 - 0.827i$
Analytic conductor: \(0.572926\)
Root analytic conductor: \(0.756919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :0),\ 0.561 - 0.827i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195720551\)
\(L(\frac12)\) \(\approx\) \(1.195720551\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \)
11 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
47 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.908512364877481454198963744331, −9.491205703723639813720828915600, −8.841004405613924494103386514785, −7.82446535985600461199480144989, −6.45175274663308295600017891949, −5.81294941476260833652931682925, −5.17632949150707308556497845418, −3.84204859307483786844571902384, −3.21071544385479668746564437921, −1.68063720069178119651450381867, 1.29128018708751784883325961845, 2.37589993944782810652193623903, 3.46689244697828483856567413030, 4.85966356415199561902565147295, 5.81946770652218343177443959665, 6.77663276719687254083327584981, 7.25134197481090866197374594357, 7.84370383878735528300799677597, 9.456433162195465427449154006183, 9.818914673755282754987293067860

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