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Properties

Label 2-52-13.5-c2-0-2
Degree $2$
Conductor $52$
Sign $0.997 - 0.0655i$
Analytic cond. $1.41689$
Root an. cond. $1.19033$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.71·3-s + (−2.77 + 2.77i)5-s + (−5.49 − 5.49i)7-s + 13.2·9-s + (6.43 + 6.43i)11-s + (−12.6 − 2.94i)13-s + (−13.1 + 13.1i)15-s − 29.1i·17-s + (−17.7 + 17.7i)19-s + (−25.9 − 25.9i)21-s + 23.5i·23-s + 9.58i·25-s + 20.1·27-s + 34.7·29-s + (29.7 − 29.7i)31-s + ⋯
L(s)  = 1  + 1.57·3-s + (−0.555 + 0.555i)5-s + (−0.785 − 0.785i)7-s + 1.47·9-s + (0.585 + 0.585i)11-s + (−0.974 − 0.226i)13-s + (−0.873 + 0.873i)15-s − 1.71i·17-s + (−0.932 + 0.932i)19-s + (−1.23 − 1.23i)21-s + 1.02i·23-s + 0.383i·25-s + 0.746·27-s + 1.19·29-s + (0.958 − 0.958i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0655i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.997 - 0.0655i$
Analytic conductor: \(1.41689\)
Root analytic conductor: \(1.19033\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1),\ 0.997 - 0.0655i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.51765 + 0.0498134i\)
\(L(\frac12)\) \(\approx\) \(1.51765 + 0.0498134i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (12.6 + 2.94i)T \)
good3 \( 1 - 4.71T + 9T^{2} \)
5 \( 1 + (2.77 - 2.77i)T - 25iT^{2} \)
7 \( 1 + (5.49 + 5.49i)T + 49iT^{2} \)
11 \( 1 + (-6.43 - 6.43i)T + 121iT^{2} \)
17 \( 1 + 29.1iT - 289T^{2} \)
19 \( 1 + (17.7 - 17.7i)T - 361iT^{2} \)
23 \( 1 - 23.5iT - 529T^{2} \)
29 \( 1 - 34.7T + 841T^{2} \)
31 \( 1 + (-29.7 + 29.7i)T - 961iT^{2} \)
37 \( 1 + (10.7 + 10.7i)T + 1.36e3iT^{2} \)
41 \( 1 + (-18.0 + 18.0i)T - 1.68e3iT^{2} \)
43 \( 1 - 32.9iT - 1.84e3T^{2} \)
47 \( 1 + (-22.9 - 22.9i)T + 2.20e3iT^{2} \)
53 \( 1 - 22.6T + 2.80e3T^{2} \)
59 \( 1 + (-41.6 - 41.6i)T + 3.48e3iT^{2} \)
61 \( 1 - 3.35T + 3.72e3T^{2} \)
67 \( 1 + (-54.3 + 54.3i)T - 4.48e3iT^{2} \)
71 \( 1 + (56.8 - 56.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (80.6 + 80.6i)T + 5.32e3iT^{2} \)
79 \( 1 + 84.3T + 6.24e3T^{2} \)
83 \( 1 + (-27.1 + 27.1i)T - 6.88e3iT^{2} \)
89 \( 1 + (-5.96 - 5.96i)T + 7.92e3iT^{2} \)
97 \( 1 + (31.9 - 31.9i)T - 9.40e3iT^{2} \)
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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

−14.99627080808201200113333323196, −14.21773482868736008388229779489, −13.27788451917714337892692212548, −11.92396111353585939969167439823, −10.13055381210869335774101451717, −9.336399976309250619214335766005, −7.74576821591320241201782089472, −7.00005969022885514979793592637, −4.10268333303747076160824076103, −2.82350274516563612157886375284, 2.66137904249043855170885474604, 4.22541089243957823784633434075, 6.59547328346276596735774514540, 8.463174038110762947161549249170, 8.731460165029927984139378745813, 10.16596556869040274532383729460, 12.16593744954975626869721863039, 12.93015925085926558385338688101, 14.24144815214891540745368002315, 15.11098067257299029358545057007

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