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Properties

Label 2-1057-1057.685-c0-0-0
Degree $2$
Conductor $1057$
Sign $0.800 + 0.598i$
Analytic cond. $0.527511$
Root an. cond. $0.726300$
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  + (−0.101 − 0.0738i)2-s + (−0.304 − 0.936i)4-s + (−0.637 + 0.770i)7-s + (−0.0770 + 0.236i)8-s + (0.968 − 0.248i)9-s + (1.60 + 0.202i)11-s + (0.121 − 0.0312i)14-s + (−0.770 + 0.560i)16-s + (−0.116 − 0.0462i)18-s + (−0.148 − 0.139i)22-s + (−0.5 − 1.53i)23-s + (0.876 − 0.481i)25-s + (0.915 + 0.362i)28-s + (−0.273 − 0.256i)29-s + 0.368·32-s + ⋯
L(s)  = 1  + (−0.101 − 0.0738i)2-s + (−0.304 − 0.936i)4-s + (−0.637 + 0.770i)7-s + (−0.0770 + 0.236i)8-s + (0.968 − 0.248i)9-s + (1.60 + 0.202i)11-s + (0.121 − 0.0312i)14-s + (−0.770 + 0.560i)16-s + (−0.116 − 0.0462i)18-s + (−0.148 − 0.139i)22-s + (−0.5 − 1.53i)23-s + (0.876 − 0.481i)25-s + (0.915 + 0.362i)28-s + (−0.273 − 0.256i)29-s + 0.368·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ 0.800 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9754721728\)
\(L(\frac12)\) \(\approx\) \(0.9754721728\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.637 - 0.770i)T \)
151 \( 1 + (0.929 + 0.368i)T \)
good2 \( 1 + (0.101 + 0.0738i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.968 + 0.248i)T^{2} \)
5 \( 1 + (-0.876 + 0.481i)T^{2} \)
11 \( 1 + (-1.60 - 0.202i)T + (0.968 + 0.248i)T^{2} \)
13 \( 1 + (0.929 + 0.368i)T^{2} \)
17 \( 1 + (0.187 - 0.982i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.273 + 0.256i)T + (0.0627 + 0.998i)T^{2} \)
31 \( 1 + (0.992 - 0.125i)T^{2} \)
37 \( 1 + (-0.303 + 1.58i)T + (-0.929 - 0.368i)T^{2} \)
41 \( 1 + (-0.535 - 0.844i)T^{2} \)
43 \( 1 + (-1.26 - 1.52i)T + (-0.187 + 0.982i)T^{2} \)
47 \( 1 + (-0.535 - 0.844i)T^{2} \)
53 \( 1 + (0.746 - 1.58i)T + (-0.637 - 0.770i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.968 - 0.248i)T^{2} \)
67 \( 1 + (1.80 + 0.462i)T + (0.876 + 0.481i)T^{2} \)
71 \( 1 + (1.23 - 1.49i)T + (-0.187 - 0.982i)T^{2} \)
73 \( 1 + (0.187 + 0.982i)T^{2} \)
79 \( 1 + (-0.00788 + 0.125i)T + (-0.992 - 0.125i)T^{2} \)
83 \( 1 + (0.425 + 0.904i)T^{2} \)
89 \( 1 + (0.425 + 0.904i)T^{2} \)
97 \( 1 + (-0.876 - 0.481i)T^{2} \)
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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

−9.930446543641039342757678763070, −9.142977931504560518651615131261, −8.870431195228862769442887114850, −7.38808086509846743737306948145, −6.28938036095248134498689123239, −6.14133057521166855818480911126, −4.66144381169650078468034831153, −4.03146001716268879932547571057, −2.47675221875620238814822247366, −1.20311959245771802153182911921, 1.43825031950044974993650038233, 3.29369596696575796590794028947, 3.87763957881042737353950454164, 4.70702889272567467284471321401, 6.19605244391198973396074645536, 7.09700834486535308664107762347, 7.46220191919350654425381299998, 8.624042708047958639908102426834, 9.388976599278467422640517621968, 9.947724325480685140889249235078

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