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Properties

Label 2-520-104.101-c1-0-14
Degree $2$
Conductor $520$
Sign $0.0796 - 0.996i$
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  + (−1.30 + 0.554i)2-s + (0.176 − 0.101i)3-s + (1.38 − 1.44i)4-s + 5-s + (−0.173 + 0.230i)6-s + (0.820 + 0.473i)7-s + (−0.999 + 2.64i)8-s + (−1.47 + 2.56i)9-s + (−1.30 + 0.554i)10-s + (0.455 + 0.788i)11-s + (0.0972 − 0.395i)12-s + (−2.46 + 2.63i)13-s + (−1.33 − 0.160i)14-s + (0.176 − 0.101i)15-s + (−0.168 − 3.99i)16-s + (−1.02 + 1.77i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.392i)2-s + (0.101 − 0.0588i)3-s + (0.692 − 0.721i)4-s + 0.447·5-s + (−0.0706 + 0.0941i)6-s + (0.310 + 0.179i)7-s + (−0.353 + 0.935i)8-s + (−0.493 + 0.854i)9-s + (−0.411 + 0.175i)10-s + (0.137 + 0.237i)11-s + (0.0280 − 0.114i)12-s + (−0.683 + 0.729i)13-s + (−0.355 − 0.0429i)14-s + (0.0455 − 0.0263i)15-s + (−0.0421 − 0.999i)16-s + (−0.248 + 0.431i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.0796 - 0.996i$
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: \((2,\ 520,\ (\ :1/2),\ 0.0796 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699677 + 0.645986i\)
\(L(\frac12)\) \(\approx\) \(0.699677 + 0.645986i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.554i)T \)
5 \( 1 - T \)
13 \( 1 + (2.46 - 2.63i)T \)
good3 \( 1 + (-0.176 + 0.101i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-0.820 - 0.473i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.455 - 0.788i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.02 - 1.77i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.975 + 1.68i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.80 - 6.58i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.35 + 4.24i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.07iT - 31T^{2} \)
37 \( 1 + (5.00 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.12 - 2.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.14 - 2.39i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 + (-4.67 + 8.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.13 - 1.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.283 + 0.491i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-8.63 - 4.98i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.67iT - 73T^{2} \)
79 \( 1 - 5.13T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + (1.09 - 0.634i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.15 + 3.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.97882022457695584469151245992, −10.03510670539988173099648595143, −9.226414074617937549347954200620, −8.493361157099738673754227071196, −7.53381779246546820518224360462, −6.74974327836501840978314972653, −5.60627502144894919236160114379, −4.78002797585200968914669903065, −2.71248203388969820265342267143, −1.64541499479438974188194397674, 0.76470986271981354576136317840, 2.46404727609633376248121839187, 3.44264558712434289879259311658, 4.98813574211627326999586478497, 6.33589991779199070785204836393, 7.09915261038516645517193852894, 8.329237694881032459937925497375, 8.830864711139020381053759031108, 9.895599838529008787999698702445, 10.41527900862204072383422763766

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