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Properties

Label 2-52-52.15-c1-0-2
Degree $2$
Conductor $52$
Sign $0.584 - 0.811i$
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  + (0.236 + 1.39i)2-s + (0.736 − 0.425i)3-s + (−1.88 + 0.659i)4-s + (−0.166 + 0.166i)5-s + (0.766 + 0.925i)6-s + (−0.684 − 2.55i)7-s + (−1.36 − 2.47i)8-s + (−1.13 + 1.97i)9-s + (−0.271 − 0.192i)10-s + (1.39 + 0.373i)11-s + (−1.10 + 1.28i)12-s + (−0.406 − 3.58i)13-s + (3.39 − 1.55i)14-s + (−0.0517 + 0.193i)15-s + (3.12 − 2.49i)16-s + (1.21 + 0.699i)17-s + ⋯
L(s)  = 1  + (0.167 + 0.985i)2-s + (0.425 − 0.245i)3-s + (−0.944 + 0.329i)4-s + (−0.0744 + 0.0744i)5-s + (0.313 + 0.377i)6-s + (−0.258 − 0.965i)7-s + (−0.483 − 0.875i)8-s + (−0.379 + 0.657i)9-s + (−0.0858 − 0.0609i)10-s + (0.419 + 0.112i)11-s + (−0.320 + 0.371i)12-s + (−0.112 − 0.993i)13-s + (0.908 − 0.416i)14-s + (−0.0133 + 0.0498i)15-s + (0.782 − 0.622i)16-s + (0.293 + 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(52\)    =    \(2^{2} \cdot 13\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(0.415222\)
Root analytic conductor: \(0.644377\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{52} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 52,\ (\ :1/2),\ 0.584 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795795 + 0.407786i\)
\(L(\frac12)\) \(\approx\) \(0.795795 + 0.407786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.236 - 1.39i)T \)
13 \( 1 + (0.406 + 3.58i)T \)
good3 \( 1 + (-0.736 + 0.425i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.166 - 0.166i)T - 5iT^{2} \)
7 \( 1 + (0.684 + 2.55i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.39 - 0.373i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.21 - 0.699i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.39 - 1.44i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.37 - 7.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.11 + 3.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.88 + 3.88i)T + 31iT^{2} \)
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 \( 1 + (-4.59 + 7.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.80 + 2.80i)T - 47iT^{2} \)
53 \( 1 + 5.94T + 53T^{2} \)
59 \( 1 + (2.20 + 8.22i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.61 + 6.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.652 + 2.43i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.5 - 2.81i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-5.05 - 5.05i)T + 73iT^{2} \)
79 \( 1 - 8.51iT - 79T^{2} \)
83 \( 1 + (-6.91 - 6.91i)T + 83iT^{2} \)
89 \( 1 + (1.71 - 6.41i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (4.00 + 14.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−15.49068755394866088124155128105, −14.58473782868769181539769390339, −13.52171100082790587779499915897, −12.82982809592734776173879372240, −10.89598545715213585371331923874, −9.444512968528746522585869738538, −8.017536786258481271195288612571, −7.19119728492743008133129686864, −5.54453422217700533134883615565, −3.71660425719795615591387877204, 2.69858647589604942391039040705, 4.37736992968356998937624781901, 6.23181125529026339797337553487, 8.790168064901698487147876266969, 9.164002338051144325984639322954, 10.74769064868172419908817977874, 12.00119942370092044758506824464, 12.72644815086084154207902814041, 14.34014571874974819723943785746, 14.83224238186560755000776268382

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