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Properties

Label 2-520-65.47-c1-0-11
Degree $2$
Conductor $520$
Sign $0.966 + 0.256i$
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  + (−2.41 + 2.41i)3-s + (2 − i)5-s − 0.828·7-s − 8.65i·9-s + (2.41 − 2.41i)11-s + (−2 − 3i)13-s + (−2.41 + 7.24i)15-s + (−1 + i)17-s + (3.24 − 3.24i)19-s + (1.99 − 1.99i)21-s + (−2.41 − 2.41i)23-s + (3 − 4i)25-s + (13.6 + 13.6i)27-s + 1.65i·29-s + (1.58 + 1.58i)31-s + ⋯
L(s)  = 1  + (−1.39 + 1.39i)3-s + (0.894 − 0.447i)5-s − 0.313·7-s − 2.88i·9-s + (0.727 − 0.727i)11-s + (−0.554 − 0.832i)13-s + (−0.623 + 1.87i)15-s + (−0.242 + 0.242i)17-s + (0.743 − 0.743i)19-s + (0.436 − 0.436i)21-s + (−0.503 − 0.503i)23-s + (0.600 − 0.800i)25-s + (2.62 + 2.62i)27-s + 0.307i·29-s + (0.284 + 0.284i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.966 + 0.256i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.966 + 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919962 - 0.120073i\)
\(L(\frac12)\) \(\approx\) \(0.919962 - 0.120073i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
13 \( 1 + (2 + 3i)T \)
good3 \( 1 + (2.41 - 2.41i)T - 3iT^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 + (-2.41 + 2.41i)T - 11iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 + (-3.24 + 3.24i)T - 19iT^{2} \)
23 \( 1 + (2.41 + 2.41i)T + 23iT^{2} \)
29 \( 1 - 1.65iT - 29T^{2} \)
31 \( 1 + (-1.58 - 1.58i)T + 31iT^{2} \)
37 \( 1 - 9.65T + 37T^{2} \)
41 \( 1 + (-6.65 - 6.65i)T + 41iT^{2} \)
43 \( 1 + (6.41 + 6.41i)T + 43iT^{2} \)
47 \( 1 + 0.828T + 47T^{2} \)
53 \( 1 + (-6.65 + 6.65i)T - 53iT^{2} \)
59 \( 1 + (7.24 + 7.24i)T + 59iT^{2} \)
61 \( 1 - 3.65T + 61T^{2} \)
67 \( 1 + 1.65iT - 67T^{2} \)
71 \( 1 + (0.757 + 0.757i)T + 71iT^{2} \)
73 \( 1 - 3.65iT - 73T^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 - 4.82T + 83T^{2} \)
89 \( 1 + (9 + 9i)T + 89iT^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.79725765171504892288091542089, −9.855988181441349025958754264096, −9.538908602438368722640988983999, −8.525114690510893681858035881810, −6.67913919018874215405997228525, −5.99472574520933268596534462921, −5.23310791451518013057886800644, −4.41228683617833658200149721017, −3.15433187164225618022850159969, −0.71772864359481187716599040987, 1.39836347030350348315002328719, 2.38753317872005031893518557362, 4.54254046752207391530510923147, 5.70691739905919341564766686757, 6.31872682671416335927989439165, 7.06448248070195933705541625218, 7.74467809924624573861435644208, 9.407975757745619808866231581739, 10.09326677639630890273692366830, 11.17341365407342554957248346849

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