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Properties

Label 2-520-65.33-c1-0-7
Degree $2$
Conductor $520$
Sign $0.279 - 0.960i$
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  + (−0.411 + 0.110i)3-s + (0.171 + 2.22i)5-s + (2.47 − 1.42i)7-s + (−2.44 + 1.40i)9-s + (0.858 + 3.20i)11-s + (1.25 − 3.38i)13-s + (−0.316 − 0.898i)15-s + (−1.43 + 5.35i)17-s + (1.51 + 0.406i)19-s + (−0.861 + 0.861i)21-s + (1.37 + 5.14i)23-s + (−4.94 + 0.766i)25-s + (1.75 − 1.75i)27-s + (−3.01 − 1.73i)29-s + (3.17 + 3.17i)31-s + ⋯
L(s)  = 1  + (−0.237 + 0.0636i)3-s + (0.0768 + 0.997i)5-s + (0.935 − 0.540i)7-s + (−0.813 + 0.469i)9-s + (0.258 + 0.965i)11-s + (0.348 − 0.937i)13-s + (−0.0817 − 0.232i)15-s + (−0.347 + 1.29i)17-s + (0.348 + 0.0933i)19-s + (−0.188 + 0.188i)21-s + (0.287 + 1.07i)23-s + (−0.988 + 0.153i)25-s + (0.337 − 0.337i)27-s + (−0.559 − 0.322i)29-s + (0.569 + 0.569i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.279 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.279 - 0.960i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ 0.279 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06566 + 0.800094i\)
\(L(\frac12)\) \(\approx\) \(1.06566 + 0.800094i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.171 - 2.22i)T \)
13 \( 1 + (-1.25 + 3.38i)T \)
good3 \( 1 + (0.411 - 0.110i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.47 + 1.42i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.858 - 3.20i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.43 - 5.35i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.51 - 0.406i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.37 - 5.14i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.01 + 1.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.17 - 3.17i)T + 31iT^{2} \)
37 \( 1 + (-2.35 - 1.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.432 - 0.115i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.84 - 1.29i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.91iT - 47T^{2} \)
53 \( 1 + (-3.95 - 3.95i)T + 53iT^{2} \)
59 \( 1 + (-2.59 + 9.68i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.43 + 7.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.29 + 2.24i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.50 + 13.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 1.21T + 73T^{2} \)
79 \( 1 + 0.932iT - 79T^{2} \)
83 \( 1 + 16.1iT - 83T^{2} \)
89 \( 1 + (10.3 - 2.76i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.81 - 3.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.92426253620670992756648179835, −10.49322891574160335731103860368, −9.427402137362874570039288983723, −8.048276695325638384761404878799, −7.63857568759059922732879651564, −6.42693063139883340950551499597, −5.52572346908437059806975340357, −4.38550054631266867106643252105, −3.16252408657571419114383743917, −1.75472545118146889070446877388, 0.862717785060357630600137781287, 2.48047284927657264485492041923, 4.08609769066455825300774909400, 5.16127104077039848111355468639, 5.83976812258480613212050409024, 6.98596482290366275148415841471, 8.421520791409636428406946781219, 8.743022296055170044943568697406, 9.523720473732805459249911354349, 11.05202004593169400311914065357

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