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Properties

Label 2-520-104.69-c1-0-38
Degree $2$
Conductor $520$
Sign $0.236 + 0.971i$
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  + (0.164 − 1.40i)2-s + (1.79 + 1.03i)3-s + (−1.94 − 0.462i)4-s − 5-s + (1.75 − 2.35i)6-s + (3.65 − 2.10i)7-s + (−0.969 + 2.65i)8-s + (0.658 + 1.14i)9-s + (−0.164 + 1.40i)10-s + (2.34 − 4.05i)11-s + (−3.02 − 2.85i)12-s + (−3.60 + 0.189i)13-s + (−2.36 − 5.48i)14-s + (−1.79 − 1.03i)15-s + (3.57 + 1.79i)16-s + (1.44 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.116 − 0.993i)2-s + (1.03 + 0.599i)3-s + (−0.972 − 0.231i)4-s − 0.447·5-s + (0.716 − 0.961i)6-s + (1.38 − 0.797i)7-s + (−0.342 + 0.939i)8-s + (0.219 + 0.380i)9-s + (−0.0520 + 0.444i)10-s + (0.705 − 1.22i)11-s + (−0.872 − 0.823i)12-s + (−0.998 + 0.0524i)13-s + (−0.631 − 1.46i)14-s + (−0.464 − 0.268i)15-s + (0.893 + 0.449i)16-s + (0.349 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.236 + 0.971i$
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: \((2,\ 520,\ (\ :1/2),\ 0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58069 - 1.24169i\)
\(L(\frac12)\) \(\approx\) \(1.58069 - 1.24169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.164 + 1.40i)T \)
5 \( 1 + T \)
13 \( 1 + (3.60 - 0.189i)T \)
good3 \( 1 + (-1.79 - 1.03i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.65 + 2.10i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.34 + 4.05i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.44 - 2.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.09 - 5.36i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.36 + 5.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.64 + 2.68i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.540iT - 31T^{2} \)
37 \( 1 + (-0.815 + 1.41i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.36 - 0.789i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.99 - 2.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.24iT - 47T^{2} \)
53 \( 1 - 10.7iT - 53T^{2} \)
59 \( 1 + (-6.90 - 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.45 - 3.72i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.52 - 9.57i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.93 - 5.15i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.13iT - 73T^{2} \)
79 \( 1 - 1.61T + 79T^{2} \)
83 \( 1 + 3.34T + 83T^{2} \)
89 \( 1 + (2.76 + 1.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.2 - 7.09i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.63915080192071895087921246010, −9.963068225141115831296842798094, −8.878885258517545813702322271973, −8.325656035545938346213757016640, −7.53835656986924000920804129674, −5.66834811119193258009147688515, −4.39284365101256594036722447381, −3.87137323456400241489416727013, −2.81105654334428395697052199666, −1.24682084709249682841399040398, 1.79368889650067509760944600728, 3.19450128928234869027592457659, 4.76626397580538026480148602963, 5.21862668094379599661123532025, 7.03685339576331675369178819396, 7.43653995082466444645552968475, 8.125426306612772650557729042919, 9.128312819851898963906718324157, 9.520449079200257559271561267291, 11.34812029996598968280292939024

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