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Properties

Label 2-520-104.101-c1-0-12
Degree $2$
Conductor $520$
Sign $0.798 - 0.602i$
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  + (−0.936 − 1.05i)2-s + (0.700 − 0.404i)3-s + (−0.245 + 1.98i)4-s − 5-s + (−1.08 − 0.363i)6-s + (3.38 + 1.95i)7-s + (2.33 − 1.59i)8-s + (−1.17 + 2.03i)9-s + (0.936 + 1.05i)10-s + (−0.223 − 0.387i)11-s + (0.630 + 1.48i)12-s + (−3.01 + 1.98i)13-s + (−1.09 − 5.41i)14-s + (−0.700 + 0.404i)15-s + (−3.87 − 0.975i)16-s + (−4.00 + 6.94i)17-s + ⋯
L(s)  = 1  + (−0.662 − 0.749i)2-s + (0.404 − 0.233i)3-s + (−0.122 + 0.992i)4-s − 0.447·5-s + (−0.442 − 0.148i)6-s + (1.27 + 0.737i)7-s + (0.824 − 0.565i)8-s + (−0.390 + 0.677i)9-s + (0.296 + 0.335i)10-s + (−0.0674 − 0.116i)11-s + (0.182 + 0.429i)12-s + (−0.835 + 0.550i)13-s + (−0.293 − 1.44i)14-s + (−0.180 + 0.104i)15-s + (−0.969 − 0.243i)16-s + (−0.972 + 1.68i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.798 - 0.602i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.798 - 0.602i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.929456 + 0.311530i\)
\(L(\frac12)\) \(\approx\) \(0.929456 + 0.311530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.936 + 1.05i)T \)
5 \( 1 + T \)
13 \( 1 + (3.01 - 1.98i)T \)
good3 \( 1 + (-0.700 + 0.404i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.38 - 1.95i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.223 + 0.387i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.00 - 6.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.861 - 1.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.570 + 0.987i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.04 + 4.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 + (-2.59 - 4.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.37 + 2.52i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.587 - 0.339i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.71iT - 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 + (4.67 - 8.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.84 + 3.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.35 - 9.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.05 - 5.22i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.118iT - 73T^{2} \)
79 \( 1 - 17.2T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-7.70 + 4.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.84 - 4.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.96678295177877004565976040554, −10.26139618414909694536028155055, −8.928446254991227496480295361799, −8.281415536984953503996108191136, −7.964334717197227269386762879328, −6.65935789110566449628648746342, −5.03710532009695412511461726179, −4.12293812148106245898870592775, −2.55173997710747352536748535804, −1.80113142926449682062276100195, 0.69494584163210744270133801393, 2.61260153065509351850543415118, 4.41973782787157175483890653159, 4.98867686768257298871893604743, 6.44234904459110498052870950549, 7.49190665356538568492936930104, 7.917767746418074018194618160653, 8.993965614441257258868738626768, 9.589197860613989273250270050643, 10.78357132284896250865813704853

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