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Properties

Label 2-520-104.101-c1-0-21
Degree $2$
Conductor $520$
Sign $0.408 - 0.912i$
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  + (1.41 − 0.0361i)2-s + (−2.77 + 1.60i)3-s + (1.99 − 0.102i)4-s + 5-s + (−3.87 + 2.36i)6-s + (1.83 + 1.06i)7-s + (2.82 − 0.216i)8-s + (3.64 − 6.31i)9-s + (1.41 − 0.0361i)10-s + (1.08 + 1.88i)11-s + (−5.38 + 3.48i)12-s + (−0.138 − 3.60i)13-s + (2.63 + 1.43i)14-s + (−2.77 + 1.60i)15-s + (3.97 − 0.407i)16-s + (−3.44 + 5.97i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0255i)2-s + (−1.60 + 0.926i)3-s + (0.998 − 0.0510i)4-s + 0.447·5-s + (−1.58 + 0.966i)6-s + (0.695 + 0.401i)7-s + (0.997 − 0.0765i)8-s + (1.21 − 2.10i)9-s + (0.447 − 0.0114i)10-s + (0.328 + 0.568i)11-s + (−1.55 + 1.00i)12-s + (−0.0385 − 0.999i)13-s + (0.705 + 0.383i)14-s + (−0.717 + 0.414i)15-s + (0.994 − 0.101i)16-s + (−0.836 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.408 - 0.912i$
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: \((2,\ 520,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60392 + 1.03887i\)
\(L(\frac12)\) \(\approx\) \(1.60392 + 1.03887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0361i)T \)
5 \( 1 - T \)
13 \( 1 + (0.138 + 3.60i)T \)
good3 \( 1 + (2.77 - 1.60i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.83 - 1.06i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.08 - 1.88i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.44 - 5.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.91 - 3.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.82 - 6.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.73 + 1.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.18iT - 31T^{2} \)
37 \( 1 + (-2.98 - 5.16i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.20 - 0.694i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.91 + 3.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.65iT - 47T^{2} \)
53 \( 1 + 8.68iT - 53T^{2} \)
59 \( 1 + (-4.70 + 8.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.984 + 0.568i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.36 - 7.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.113 - 0.0655i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 3.75iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (-11.9 + 6.89i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.694 - 0.400i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.33521552723940323353287311888, −10.31836500760293413959506495788, −9.866370606117120000299570847472, −8.287576067553610957270472771307, −6.86256690928652443756448637075, −5.99349772912387576106233753578, −5.41136870771025240743972525159, −4.60133243247408464192148651683, −3.67088096509626606233666429729, −1.70933462277398731718714744397, 1.13420540021847292407329481102, 2.43973080490837302899662171258, 4.63310247181587119922070155702, 4.90955825187276654290819855878, 6.18230956489290559517201579179, 6.76610689811139623638332370579, 7.34793509026718400454116334498, 8.832180528683623896362760018171, 10.47346351703949479980330432415, 11.14205118014245319640860163496

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