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Properties

Label 2-51-3.2-c6-0-25
Degree $2$
Conductor $51$
Sign $0.483 + 0.875i$
Analytic cond. $11.7327$
Root an. cond. $3.42531$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.9i·2-s + (23.6 − 13.0i)3-s − 131.·4-s − 76.5i·5-s + (182. + 330. i)6-s − 548.·7-s − 939. i·8-s + (388. − 616. i)9-s + 1.06e3·10-s − 1.54e3i·11-s + (−3.10e3 + 1.71e3i)12-s − 3.44e3·13-s − 7.66e3i·14-s + (−998. − 1.80e3i)15-s + 4.72e3·16-s − 1.19e3i·17-s + ⋯
L(s)  = 1  + 1.74i·2-s + (0.875 − 0.483i)3-s − 2.05·4-s − 0.612i·5-s + (0.844 + 1.52i)6-s − 1.60·7-s − 1.83i·8-s + (0.532 − 0.846i)9-s + 1.06·10-s − 1.15i·11-s + (−1.79 + 0.990i)12-s − 1.56·13-s − 2.79i·14-s + (−0.295 − 0.535i)15-s + 1.15·16-s − 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.483 + 0.875i$
Analytic conductor: \(11.7327\)
Root analytic conductor: \(3.42531\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{51} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :3),\ 0.483 + 0.875i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.541156 - 0.319387i\)
\(L(\frac12)\) \(\approx\) \(0.541156 - 0.319387i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-23.6 + 13.0i)T \)
17 \( 1 + 1.19e3iT \)
good2 \( 1 - 13.9iT - 64T^{2} \)
5 \( 1 + 76.5iT - 1.56e4T^{2} \)
7 \( 1 + 548.T + 1.17e5T^{2} \)
11 \( 1 + 1.54e3iT - 1.77e6T^{2} \)
13 \( 1 + 3.44e3T + 4.82e6T^{2} \)
19 \( 1 - 3.75e3T + 4.70e7T^{2} \)
23 \( 1 - 1.57e4iT - 1.48e8T^{2} \)
29 \( 1 - 1.37e4iT - 5.94e8T^{2} \)
31 \( 1 + 4.37e4T + 8.87e8T^{2} \)
37 \( 1 + 1.10e4T + 2.56e9T^{2} \)
41 \( 1 + 2.98e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.18e4T + 6.32e9T^{2} \)
47 \( 1 + 2.44e4iT - 1.07e10T^{2} \)
53 \( 1 + 1.64e5iT - 2.21e10T^{2} \)
59 \( 1 + 8.15e4iT - 4.21e10T^{2} \)
61 \( 1 + 4.30e5T + 5.15e10T^{2} \)
67 \( 1 + 6.41e3T + 9.04e10T^{2} \)
71 \( 1 + 3.74e5iT - 1.28e11T^{2} \)
73 \( 1 - 4.68e5T + 1.51e11T^{2} \)
79 \( 1 + 7.27e4T + 2.43e11T^{2} \)
83 \( 1 - 8.21e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.67e5iT - 4.96e11T^{2} \)
97 \( 1 + 9.31e5T + 8.32e11T^{2} \)
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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

−14.16240123005180851897589353536, −13.32672294921642420727648879438, −12.46110486311187144344843387077, −9.568835586083786566024847951630, −8.991426010886817187937782255016, −7.63182047377603294363303256548, −6.73478733353558357672993758817, −5.37881859144461047292847579639, −3.37659593993529699307039976379, −0.23717864379384447842185035480, 2.33153181746630056007258404911, 3.16319969432323171925103031665, 4.51673372308967820012274098792, 7.18626512688599366024072463254, 9.223596823861684709561195642671, 9.875506242436834135550903275608, 10.56604140470609204525602983437, 12.33663011039175065649277028506, 12.89970473136282319580942455079, 14.20675029626629769123709196755

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