Properties

Label 2-1040-13.4-c1-0-24
Degree $2$
Conductor $1040$
Sign $-0.972 + 0.231i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.268 + 0.465i)3-s i·5-s + (0.331 − 0.191i)7-s + (1.35 + 2.34i)9-s + (−5.66 − 3.27i)11-s + (−2.44 − 2.65i)13-s + (0.465 + 0.268i)15-s + (0.174 + 0.301i)17-s + (−3.55 + 2.05i)19-s + 0.205i·21-s + (−3.36 + 5.82i)23-s − 25-s − 3.06·27-s + (1.91 − 3.32i)29-s + 1.67i·31-s + ⋯
L(s)  = 1  + (−0.155 + 0.268i)3-s − 0.447i·5-s + (0.125 − 0.0723i)7-s + (0.451 + 0.782i)9-s + (−1.70 − 0.986i)11-s + (−0.677 − 0.735i)13-s + (0.120 + 0.0693i)15-s + (0.0422 + 0.0732i)17-s + (−0.815 + 0.470i)19-s + 0.0449i·21-s + (−0.701 + 1.21i)23-s − 0.200·25-s − 0.590·27-s + (0.356 − 0.617i)29-s + 0.301i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.972 + 0.231i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ -0.972 + 0.231i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1603377528\)
\(L(\frac12)\) \(\approx\) \(0.1603377528\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (2.44 + 2.65i)T \)
good3 \( 1 + (0.268 - 0.465i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.331 + 0.191i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.66 + 3.27i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.174 - 0.301i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.55 - 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.36 - 5.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.91 + 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.67iT - 31T^{2} \)
37 \( 1 + (-0.963 - 0.556i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.17 + 3.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.87 + 4.98i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.72iT - 47T^{2} \)
53 \( 1 + 7.18T + 53T^{2} \)
59 \( 1 + (-3.85 + 2.22i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.19 - 9.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.01 + 1.74i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.93 - 1.69i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.04iT - 73T^{2} \)
79 \( 1 + 10.7T + 79T^{2} \)
83 \( 1 - 8.27iT - 83T^{2} \)
89 \( 1 + (15.5 + 8.97i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.14 + 0.661i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.845698766613489223287305175124, −8.418798509206239205229054526513, −8.056092974707672414227716452993, −7.23553825576206206437355148961, −5.73300162791103516397018860231, −5.32633759759153374689278501430, −4.37692448094327530816065889007, −3.13622909776747023176204562374, −1.94902640714682182589886758903, −0.06730994294964929536256026661, 1.95457308790070411542983616614, 2.85663224302850756463069232352, 4.33039256159540066188027444684, 4.99426503483521183657563845549, 6.29882834694920970256895470007, 6.92105056112210043529104464597, 7.69556419129544801815987986941, 8.568316864769988579161090149285, 9.770560697293753902409443914761, 10.12807562299318515487276487797

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