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Properties

Label 2-1057-1057.727-c0-0-0
Degree $2$
Conductor $1057$
Sign $-0.919 - 0.393i$
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  + (−0.263 + 0.809i)2-s + (0.222 + 0.161i)4-s + (−0.992 + 0.125i)7-s + (−0.878 + 0.638i)8-s + (−0.187 + 0.982i)9-s + (−0.393 + 0.476i)11-s + (0.159 − 0.836i)14-s + (−0.200 − 0.617i)16-s + (−0.746 − 0.410i)18-s + (−0.282 − 0.444i)22-s + (−0.5 − 0.363i)23-s + (−0.929 − 0.368i)25-s + (−0.240 − 0.132i)28-s + (1.03 + 1.63i)29-s − 0.532·32-s + ⋯
L(s)  = 1  + (−0.263 + 0.809i)2-s + (0.222 + 0.161i)4-s + (−0.992 + 0.125i)7-s + (−0.878 + 0.638i)8-s + (−0.187 + 0.982i)9-s + (−0.393 + 0.476i)11-s + (0.159 − 0.836i)14-s + (−0.200 − 0.617i)16-s + (−0.746 − 0.410i)18-s + (−0.282 − 0.444i)22-s + (−0.5 − 0.363i)23-s + (−0.929 − 0.368i)25-s + (−0.240 − 0.132i)28-s + (1.03 + 1.63i)29-s − 0.532·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.919 - 0.393i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ -0.919 - 0.393i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7057825435\)
\(L(\frac12)\) \(\approx\) \(0.7057825435\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.992 - 0.125i)T \)
151 \( 1 + (-0.876 - 0.481i)T \)
good2 \( 1 + (0.263 - 0.809i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.187 - 0.982i)T^{2} \)
5 \( 1 + (0.929 + 0.368i)T^{2} \)
11 \( 1 + (0.393 - 0.476i)T + (-0.187 - 0.982i)T^{2} \)
13 \( 1 + (-0.876 - 0.481i)T^{2} \)
17 \( 1 + (-0.968 - 0.248i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (-1.03 - 1.63i)T + (-0.425 + 0.904i)T^{2} \)
31 \( 1 + (0.637 + 0.770i)T^{2} \)
37 \( 1 + (-0.598 - 0.153i)T + (0.876 + 0.481i)T^{2} \)
41 \( 1 + (-0.728 + 0.684i)T^{2} \)
43 \( 1 + (-1.26 - 0.159i)T + (0.968 + 0.248i)T^{2} \)
47 \( 1 + (-0.728 + 0.684i)T^{2} \)
53 \( 1 + (0.116 - 1.85i)T + (-0.992 - 0.125i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.187 + 0.982i)T^{2} \)
67 \( 1 + (0.328 + 1.72i)T + (-0.929 + 0.368i)T^{2} \)
71 \( 1 + (-0.371 + 0.0469i)T + (0.968 - 0.248i)T^{2} \)
73 \( 1 + (-0.968 + 0.248i)T^{2} \)
79 \( 1 + (-0.362 - 0.770i)T + (-0.637 + 0.770i)T^{2} \)
83 \( 1 + (-0.0627 - 0.998i)T^{2} \)
89 \( 1 + (-0.0627 - 0.998i)T^{2} \)
97 \( 1 + (0.929 - 0.368i)T^{2} \)
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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

−10.40061853430426687954350506083, −9.500331657443333237849016365373, −8.641503519427828341717193714475, −7.85213449022990472315430823288, −7.21085172464192244096699009967, −6.31933745056198202502279598990, −5.63292127105721827755616996069, −4.53845459724914002758370653124, −3.11560303818866580096957882915, −2.26922885970598280870882157539, 0.66637044363566623125739278889, 2.33169864192583890746400198383, 3.26037873957950143680360849650, 4.04019673320740624585919291746, 5.80214350353070010552164006657, 6.19829272626672487534294739140, 7.14474260532738360616388224524, 8.285329608723632070447406423643, 9.318863790611459382958592171955, 9.790817396807225784443702020412

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