Properties

Label 2-520-5.4-c1-0-0
Degree $2$
Conductor $520$
Sign $-0.736 + 0.676i$
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  + 3.08i·3-s + (−1.64 + 1.51i)5-s − 3.54i·7-s − 6.54·9-s − 3.86·11-s + i·13-s + (−4.67 − 5.08i)15-s − 1.29i·17-s − 3.69·19-s + 10.9·21-s + 8.38i·23-s + (0.423 − 4.98i)25-s − 10.9i·27-s + 7.53·29-s − 8.36·31-s + ⋯
L(s)  = 1  + 1.78i·3-s + (−0.736 + 0.676i)5-s − 1.33i·7-s − 2.18·9-s − 1.16·11-s + 0.277i·13-s + (−1.20 − 1.31i)15-s − 0.313i·17-s − 0.846·19-s + 2.38·21-s + 1.74i·23-s + (0.0847 − 0.996i)25-s − 2.10i·27-s + 1.39·29-s − 1.50·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.736 + 0.676i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147513 - 0.378663i\)
\(L(\frac12)\) \(\approx\) \(0.147513 - 0.378663i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 - 1.51i)T \)
13 \( 1 - iT \)
good3 \( 1 - 3.08iT - 3T^{2} \)
7 \( 1 + 3.54iT - 7T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
17 \( 1 + 1.29iT - 17T^{2} \)
19 \( 1 + 3.69T + 19T^{2} \)
23 \( 1 - 8.38iT - 23T^{2} \)
29 \( 1 - 7.53T + 29T^{2} \)
31 \( 1 + 8.36T + 31T^{2} \)
37 \( 1 - 1.04iT - 37T^{2} \)
41 \( 1 + 2.97T + 41T^{2} \)
43 \( 1 + 4.96iT - 43T^{2} \)
47 \( 1 + 0.459iT - 47T^{2} \)
53 \( 1 - 12.2iT - 53T^{2} \)
59 \( 1 + 3.28T + 59T^{2} \)
61 \( 1 - 0.582T + 61T^{2} \)
67 \( 1 - 8.15iT - 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 3.54iT - 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 2.90iT - 97T^{2} \)
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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

−10.99764430616836365946630750167, −10.51880352298194124918516614078, −10.01222667408581961513687034576, −8.914307518504840391876885412870, −7.85641002371183854406500194355, −7.04064376699320730914880720804, −5.57844233235522400162999467688, −4.51695302215487858612883110106, −3.84931034237423181685216741008, −2.95372565690340562798611614995, 0.22525753530784877543359656574, 1.96481019865096816846261115425, 2.91711355419428892569131487728, 4.85784669382569397936431743426, 5.81801997613668124452242905192, 6.68594608636861426835314650258, 7.81235787796033703281231873796, 8.364487825962576329544900925479, 8.888052874849821893771666647069, 10.57325735119090312088270670592

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