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Properties

Label 2-520-104.101-c1-0-17
Degree $2$
Conductor $520$
Sign $0.806 - 0.590i$
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  + (−1.13 − 0.844i)2-s + (−1.79 + 1.03i)3-s + (0.572 + 1.91i)4-s + 5-s + (2.91 + 0.341i)6-s + (3.65 + 2.10i)7-s + (0.969 − 2.65i)8-s + (0.658 − 1.14i)9-s + (−1.13 − 0.844i)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 + (−3.34 + 2.19i)16-s + (1.44 − 2.49i)17-s + ⋯
L(s)  = 1  + (−0.801 − 0.597i)2-s + (−1.03 + 0.599i)3-s + (0.286 + 0.958i)4-s + 0.447·5-s + (1.19 + 0.139i)6-s + (1.38 + 0.797i)7-s + (0.342 − 0.939i)8-s + (0.219 − 0.380i)9-s + (−0.358 − 0.267i)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 + (−0.836 + 0.548i)16-s + (0.349 − 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 - 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.806 - 0.590i$
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: \((2,\ 520,\ (\ :1/2),\ 0.806 - 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796381 + 0.260381i\)
\(L(\frac12)\) \(\approx\) \(0.796381 + 0.260381i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.844i)T \)
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

−10.86627605005663370706689251218, −10.45787369327991265725909781866, −9.241408818864349201915985013040, −8.429008603942203539794412099271, −7.74588590102519224213577279417, −6.01421454827691421676884965076, −5.51478778382515855011436974036, −4.29108469513602712416664195519, −2.78417040457083400144545837934, −1.28739766371890923452365242269, 0.882103951370454201762910237446, 2.00470521090412996144851018655, 4.66518865462716043984814554871, 5.26703748379984168530997338460, 6.51331477354060702325042643280, 6.98686164863124459060446955017, 8.026225366806255322250961459473, 8.769090280003424979757663586119, 10.12339430246743244909521323978, 10.86122329163413485650858103004

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