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Properties

Label 2-52-52.31-c1-0-0
Degree $2$
Conductor $52$
Sign $0.707 - 0.707i$
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  + (−1.13 − 0.842i)2-s + 2.79i·3-s + (0.579 + 1.91i)4-s + (−0.707 + 0.707i)5-s + (2.35 − 3.17i)6-s + (1.97 − 1.97i)7-s + (0.955 − 2.66i)8-s − 4.82·9-s + (1.39 − 0.207i)10-s + (−1.15 + 1.15i)11-s + (−5.35 + 1.62i)12-s + (3.53 − 0.707i)13-s + (−3.91 + 0.579i)14-s + (−1.97 − 1.97i)15-s + (−3.32 + 2.21i)16-s − 5.82i·17-s + ⋯
L(s)  = 1  + (−0.803 − 0.595i)2-s + 1.61i·3-s + (0.289 + 0.957i)4-s + (−0.316 + 0.316i)5-s + (0.962 − 1.29i)6-s + (0.747 − 0.747i)7-s + (0.337 − 0.941i)8-s − 1.60·9-s + (0.442 − 0.0654i)10-s + (−0.349 + 0.349i)11-s + (−1.54 + 0.468i)12-s + (0.980 − 0.196i)13-s + (−1.04 + 0.154i)14-s + (−0.510 − 0.510i)15-s + (−0.832 + 0.554i)16-s − 1.41i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558694 + 0.231402i\)
\(L(\frac12)\) \(\approx\) \(0.558694 + 0.231402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.842i)T \)
13 \( 1 + (-3.53 + 0.707i)T \)
good3 \( 1 - 2.79iT - 3T^{2} \)
5 \( 1 + (0.707 - 0.707i)T - 5iT^{2} \)
7 \( 1 + (-1.97 + 1.97i)T - 7iT^{2} \)
11 \( 1 + (1.15 - 1.15i)T - 11iT^{2} \)
17 \( 1 + 5.82iT - 17T^{2} \)
19 \( 1 + (1.63 + 1.63i)T + 19iT^{2} \)
23 \( 1 - 3.95T + 23T^{2} \)
29 \( 1 + 0.585T + 29T^{2} \)
31 \( 1 + (3.95 + 3.95i)T + 31iT^{2} \)
37 \( 1 + (1.87 + 1.87i)T + 37iT^{2} \)
41 \( 1 + (4.24 - 4.24i)T - 41iT^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + (1.97 - 1.97i)T - 47iT^{2} \)
53 \( 1 + 0.242T + 53T^{2} \)
59 \( 1 + (-2.31 + 2.31i)T - 59iT^{2} \)
61 \( 1 - 0.828T + 61T^{2} \)
67 \( 1 + (-2.79 - 2.79i)T + 67iT^{2} \)
71 \( 1 + (-5.25 - 5.25i)T + 71iT^{2} \)
73 \( 1 + (9.65 + 9.65i)T + 73iT^{2} \)
79 \( 1 + 9.55iT - 79T^{2} \)
83 \( 1 + (6.75 + 6.75i)T + 83iT^{2} \)
89 \( 1 + (-6.24 - 6.24i)T + 89iT^{2} \)
97 \( 1 + (-0.414 + 0.414i)T - 97iT^{2} \)
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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

−15.81172850239970504098096915052, −14.86521624068836238588465082578, −13.35711605343277017747320862931, −11.35664389447820313279566670215, −10.95778230298813705896152440363, −9.879280893817414068181893611348, −8.792299089336748476949028249436, −7.38328416295941621787530572225, −4.74311592753689110256564057429, −3.40208734862294337849673154420, 1.69372287444388214897323140660, 5.63065274842333399980524672814, 6.77488662153358119912832670336, 8.302942632119783869770863657934, 8.490251969567905526584384671148, 10.79959074914602856546581676979, 11.91887730602729011580218850898, 13.07532846473438037712756132113, 14.29718934704464108341641204207, 15.38277875574444203212634444317

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