Properties

Label 2-1057-1057.482-c0-0-0
Degree $2$
Conductor $1057$
Sign $0.730 - 0.682i$
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  + (1.03 + 0.749i)2-s + (0.193 + 0.594i)4-s + (0.968 − 0.248i)7-s + (0.147 − 0.454i)8-s + (−0.929 − 0.368i)9-s + (0.303 + 1.58i)11-s + (1.18 + 0.469i)14-s + (0.998 − 0.725i)16-s + (−0.683 − 1.07i)18-s + (−0.878 + 1.86i)22-s + (−0.5 − 1.53i)23-s + (0.728 + 0.684i)25-s + (0.335 + 0.527i)28-s + (−0.746 + 1.58i)29-s + 1.09·32-s + ⋯
L(s)  = 1  + (1.03 + 0.749i)2-s + (0.193 + 0.594i)4-s + (0.968 − 0.248i)7-s + (0.147 − 0.454i)8-s + (−0.929 − 0.368i)9-s + (0.303 + 1.58i)11-s + (1.18 + 0.469i)14-s + (0.998 − 0.725i)16-s + (−0.683 − 1.07i)18-s + (−0.878 + 1.86i)22-s + (−0.5 − 1.53i)23-s + (0.728 + 0.684i)25-s + (0.335 + 0.527i)28-s + (−0.746 + 1.58i)29-s + 1.09·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $0.730 - 0.682i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ 0.730 - 0.682i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.792811304\)
\(L(\frac12)\) \(\approx\) \(1.792811304\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.968 + 0.248i)T \)
151 \( 1 + (-0.535 - 0.844i)T \)
good2 \( 1 + (-1.03 - 0.749i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.929 + 0.368i)T^{2} \)
5 \( 1 + (-0.728 - 0.684i)T^{2} \)
11 \( 1 + (-0.303 - 1.58i)T + (-0.929 + 0.368i)T^{2} \)
13 \( 1 + (-0.535 - 0.844i)T^{2} \)
17 \( 1 + (-0.876 - 0.481i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.746 - 1.58i)T + (-0.637 - 0.770i)T^{2} \)
31 \( 1 + (0.187 - 0.982i)T^{2} \)
37 \( 1 + (1.41 + 0.779i)T + (0.535 + 0.844i)T^{2} \)
41 \( 1 + (-0.0627 + 0.998i)T^{2} \)
43 \( 1 + (0.362 + 0.0931i)T + (0.876 + 0.481i)T^{2} \)
47 \( 1 + (-0.0627 + 0.998i)T^{2} \)
53 \( 1 + (1.44 + 0.182i)T + (0.968 + 0.248i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.929 - 0.368i)T^{2} \)
67 \( 1 + (0.996 - 0.394i)T + (0.728 - 0.684i)T^{2} \)
71 \( 1 + (1.80 - 0.462i)T + (0.876 - 0.481i)T^{2} \)
73 \( 1 + (-0.876 + 0.481i)T^{2} \)
79 \( 1 + (-0.812 + 0.982i)T + (-0.187 - 0.982i)T^{2} \)
83 \( 1 + (0.992 - 0.125i)T^{2} \)
89 \( 1 + (0.992 - 0.125i)T^{2} \)
97 \( 1 + (-0.728 + 0.684i)T^{2} \)
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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

−10.31554443863533871010786721570, −9.238540318527396444673710646152, −8.433276217415809060412333559350, −7.30638167678014518522120216334, −6.88711459083357094368101085703, −5.81039539682693791663515728279, −4.96271588866128076580034942663, −4.40932689217044270588319070799, −3.29618013203198504633119371970, −1.74918707017431859653826619995, 1.71119226171164130193883083888, 2.86582613403964194870988684839, 3.64871643043555347740734807739, 4.74673936836858477681271072591, 5.55651728167529901580302316645, 6.11884428827508024746900572254, 7.79744427903337640853103957112, 8.302455292539954491036401402267, 9.076450639159873090079896247395, 10.43932100353241955425269895044

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