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Properties

Label 2-520-13.12-c1-0-9
Degree $2$
Conductor $520$
Sign $0.996 + 0.0862i$
Analytic cond. $4.15222$
Root an. cond. $2.03769$
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  + 2.21·3-s − i·5-s + 2.90i·7-s + 1.90·9-s − 5.73i·11-s + (3.59 + 0.311i)13-s − 2.21i·15-s + 5.80·17-s + 5.11i·19-s + 6.42i·21-s + 5.59·23-s − 25-s − 2.42·27-s − 8.57·29-s + 1.44i·31-s + ⋯
L(s)  = 1  + 1.27·3-s − 0.447i·5-s + 1.09i·7-s + 0.634·9-s − 1.73i·11-s + (0.996 + 0.0862i)13-s − 0.571i·15-s + 1.40·17-s + 1.17i·19-s + 1.40i·21-s + 1.16·23-s − 0.200·25-s − 0.467·27-s − 1.59·29-s + 0.259i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.996 + 0.0862i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.996 + 0.0862i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21894 - 0.0959105i\)
\(L(\frac12)\) \(\approx\) \(2.21894 - 0.0959105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.59 - 0.311i)T \)
good3 \( 1 - 2.21T + 3T^{2} \)
7 \( 1 - 2.90iT - 7T^{2} \)
11 \( 1 + 5.73iT - 11T^{2} \)
17 \( 1 - 5.80T + 17T^{2} \)
19 \( 1 - 5.11iT - 19T^{2} \)
23 \( 1 - 5.59T + 23T^{2} \)
29 \( 1 + 8.57T + 29T^{2} \)
31 \( 1 - 1.44iT - 31T^{2} \)
37 \( 1 + 0.474iT - 37T^{2} \)
41 \( 1 + 9.47iT - 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 - 8.70iT - 47T^{2} \)
53 \( 1 + 9.47T + 53T^{2} \)
59 \( 1 + 0.688iT - 59T^{2} \)
61 \( 1 + 3.85T + 61T^{2} \)
67 \( 1 - 5.52iT - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 + 0.561T + 79T^{2} \)
83 \( 1 + 7.03iT - 83T^{2} \)
89 \( 1 - 1.80iT - 89T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−10.90001259014701149126930340698, −9.633188474801713344708838400541, −8.807328771523395874190286582227, −8.474274680161090344185381098798, −7.65686183322127789906197311912, −5.98934407387314998510187336801, −5.46605726124465745295766604543, −3.60204928733804626900859340499, −3.11991960616155416087393060926, −1.55946052307349935851148426919, 1.63445413674556495583183579738, 3.05230212580548770520633155933, 3.84729223034356995790342844812, 5.02304279912274639161494081845, 6.68845892005823649756374274696, 7.41329026753843705724165188297, 8.043922204562300660408763811238, 9.253542635855182769837828820134, 9.825246541528618074058629300851, 10.72347152688428224937321257876

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