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Properties

Label 2-52-52.7-c1-0-3
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: $\chi_{52} (7, \cdot )$
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

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

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