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Properties

Label 2-1169-1169.333-c0-0-11
Degree $2$
Conductor $1169$
Sign $0.126 + 0.991i$
Analytic cond. $0.583406$
Root an. cond. $0.763810$
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  + (0.142 + 0.246i)2-s + (0.888 − 1.53i)3-s + (0.459 − 0.795i)4-s + 0.505·6-s + (−0.327 + 0.945i)7-s + 0.546·8-s + (−1.08 − 1.87i)9-s + (−0.235 + 0.408i)11-s + (−0.816 − 1.41i)12-s + (−0.279 + 0.0538i)14-s + (−0.381 − 0.661i)16-s + (0.307 − 0.532i)18-s + (−0.415 − 0.719i)19-s + (1.16 + 1.34i)21-s − 0.134·22-s + ⋯
L(s)  = 1  + (0.142 + 0.246i)2-s + (0.888 − 1.53i)3-s + (0.459 − 0.795i)4-s + 0.505·6-s + (−0.327 + 0.945i)7-s + 0.546·8-s + (−1.08 − 1.87i)9-s + (−0.235 + 0.408i)11-s + (−0.816 − 1.41i)12-s + (−0.279 + 0.0538i)14-s + (−0.381 − 0.661i)16-s + (0.307 − 0.532i)18-s + (−0.415 − 0.719i)19-s + (1.16 + 1.34i)21-s − 0.134·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1169\)    =    \(7 \cdot 167\)
Sign: $0.126 + 0.991i$
Analytic conductor: \(0.583406\)
Root analytic conductor: \(0.763810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1169} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1169,\ (\ :0),\ 0.126 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564592552\)
\(L(\frac12)\) \(\approx\) \(1.564592552\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.327 - 0.945i)T \)
167 \( 1 - T \)
good2 \( 1 + (-0.142 - 0.246i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.888 + 1.53i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.415 + 0.719i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.96T + T^{2} \)
31 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + 1.91T + T^{2} \)
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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

−9.499364776009116941039578492619, −8.888163035666382858468767467967, −8.031273766280608701612104649110, −7.16465441243921366639444907429, −6.60714085820706156014817615662, −5.88670148443439573127438154994, −4.86909684653969290826309226816, −3.08168684566246225732239295617, −2.33958912540938755610289945719, −1.39433189456944800881750172532, 2.33732670246099255852485687764, 3.21943275166405864644589541436, 4.01876026019531666996754179957, 4.44324850439450437742088608684, 5.83919058268031425164601626470, 7.04411725668101526202777732811, 8.036549687087671290171123135913, 8.447283308040258817851325524880, 9.475092062501936562652712910876, 10.36910773094498496769992102911

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