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Properties

Label 2-52-52.7-c1-0-1
Degree $2$
Conductor $52$
Sign $0.396 - 0.918i$
Analytic cond. $0.415222$
Root an. cond. $0.644377$
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  + (−0.526 + 1.31i)2-s + (2.16 + 1.25i)3-s + (−1.44 − 1.38i)4-s + (−2.19 − 2.19i)5-s + (−2.78 + 2.18i)6-s + (−0.152 + 0.569i)7-s + (2.57 − 1.16i)8-s + (1.63 + 2.83i)9-s + (4.04 − 1.72i)10-s + (−2.85 + 0.764i)11-s + (−1.40 − 4.80i)12-s + (2.37 − 2.71i)13-s + (−0.667 − 0.500i)14-s + (−2.01 − 7.52i)15-s + (0.179 + 3.99i)16-s + (−2.30 + 1.33i)17-s + ⋯
L(s)  = 1  + (−0.372 + 0.928i)2-s + (1.25 + 0.723i)3-s + (−0.722 − 0.691i)4-s + (−0.983 − 0.983i)5-s + (−1.13 + 0.893i)6-s + (−0.0576 + 0.215i)7-s + (0.910 − 0.413i)8-s + (0.546 + 0.946i)9-s + (1.27 − 0.546i)10-s + (−0.860 + 0.230i)11-s + (−0.405 − 1.38i)12-s + (0.657 − 0.753i)13-s + (−0.178 − 0.133i)14-s + (−0.520 − 1.94i)15-s + (0.0449 + 0.998i)16-s + (−0.560 + 0.323i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.396 - 0.918i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.396 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688109 + 0.452459i\)
\(L(\frac12)\) \(\approx\) \(0.688109 + 0.452459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.526 - 1.31i)T \)
13 \( 1 + (-2.37 + 2.71i)T \)
good3 \( 1 + (-2.16 - 1.25i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (2.19 + 2.19i)T + 5iT^{2} \)
7 \( 1 + (0.152 - 0.569i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.85 - 0.764i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.30 - 1.33i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 0.835i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.03 - 1.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.621 - 1.07i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.34 - 6.34i)T - 31iT^{2} \)
37 \( 1 + (0.133 + 0.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.59 + 1.5i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.60 + 6.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.16 - 3.16i)T + 47iT^{2} \)
53 \( 1 - 3.67T + 53T^{2} \)
59 \( 1 + (-1.82 + 6.80i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.97 + 6.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.92 - 7.18i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.98 - 0.530i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.0440 + 0.0440i)T - 73iT^{2} \)
79 \( 1 - 2.73iT - 79T^{2} \)
83 \( 1 + (-10.1 + 10.1i)T - 83iT^{2} \)
89 \( 1 + (-2.14 - 8.01i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.22 - 4.58i)T + (-84.0 - 48.5i)T^{2} \)
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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

−15.76911369042123987095611222204, −15.00064008094007994302726448429, −13.74902425811294494307865445567, −12.64823231216606508618886116510, −10.59576910880828179547086758222, −9.234488243823770805419356174672, −8.446074000671785654891709169571, −7.64273346282807461895755750664, −5.25468222936889579894861396139, −3.83345855683635656007752329412, 2.57555361163562246479293078353, 3.81537264573178636856392011621, 7.21726432500879692583106424966, 8.004099666798507673025801503897, 9.183438406606072508903695021825, 10.76297947868636936510574432952, 11.68519828343756302238397691431, 13.14964908535024505624154558080, 13.85564227480798002665512605459, 14.99316516123861635990273209900

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