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Properties

Label 2-520-104.69-c1-0-40
Degree $2$
Conductor $520$
Sign $-0.393 + 0.919i$
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  + (−1.37 + 0.314i)2-s + (1.00 + 0.580i)3-s + (1.80 − 0.866i)4-s − 5-s + (−1.56 − 0.484i)6-s + (−2.74 + 1.58i)7-s + (−2.21 + 1.76i)8-s + (−0.826 − 1.43i)9-s + (1.37 − 0.314i)10-s + (1.23 − 2.13i)11-s + (2.31 + 0.175i)12-s + (−3.60 − 0.150i)13-s + (3.28 − 3.04i)14-s + (−1.00 − 0.580i)15-s + (2.49 − 3.12i)16-s + (0.369 + 0.639i)17-s + ⋯
L(s)  = 1  + (−0.975 + 0.222i)2-s + (0.580 + 0.335i)3-s + (0.901 − 0.433i)4-s − 0.447·5-s + (−0.640 − 0.197i)6-s + (−1.03 + 0.598i)7-s + (−0.782 + 0.622i)8-s + (−0.275 − 0.477i)9-s + (0.436 − 0.0993i)10-s + (0.371 − 0.643i)11-s + (0.668 + 0.0506i)12-s + (−0.999 − 0.0416i)13-s + (0.877 − 0.813i)14-s + (−0.259 − 0.149i)15-s + (0.624 − 0.780i)16-s + (0.0895 + 0.155i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $-0.393 + 0.919i$
Analytic conductor: \(4.15222\)
Root analytic conductor: \(2.03769\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 520,\ (\ :1/2),\ -0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188612 - 0.285840i\)
\(L(\frac12)\) \(\approx\) \(0.188612 - 0.285840i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.314i)T \)
5 \( 1 + T \)
13 \( 1 + (3.60 + 0.150i)T \)
good3 \( 1 + (-1.00 - 0.580i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.74 - 1.58i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.23 + 2.13i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.369 - 0.639i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.31 + 7.47i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.88 + 4.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.59 + 0.919i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.05iT - 31T^{2} \)
37 \( 1 + (1.35 - 2.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.165 - 0.0952i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.43 - 4.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 8.23iT - 47T^{2} \)
53 \( 1 + 4.18iT - 53T^{2} \)
59 \( 1 + (5.72 + 9.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.03 + 2.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 2.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.24 - 1.87i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 + 1.79T + 83T^{2} \)
89 \( 1 + (-12.1 - 7.04i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.8 - 8.56i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34537044642805908278720904578, −9.453817875088727568639757195770, −8.843466854780396349829276609671, −8.311065559549145556198772687718, −6.83412052557288632239747765068, −6.46331723404372444381385200953, −4.99614085578644346882732364950, −3.32347536429935554109786659883, −2.57988806567746090943293200224, −0.23850039913141593039634511011, 1.81947966037899846799081034255, 3.03836663231841410039581612260, 4.08270555067138177090724052583, 5.90666351496517936681506198735, 7.15886690345862332931027653983, 7.51537772525961006027860131916, 8.459086991544363711731775308859, 9.490231794023385946913093569860, 10.04012227086343411267584201556, 10.96051751544935980408365514137

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