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Properties

Label 2-51-3.2-c6-0-28
Degree $2$
Conductor $51$
Sign $-0.950 - 0.311i$
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  − 7.50i·2-s + (8.40 − 25.6i)3-s + 7.66·4-s + 24.7i·5-s + (−192. − 63.1i)6-s − 318.·7-s − 537. i·8-s + (−587. − 431. i)9-s + 185.·10-s − 711. i·11-s + (64.4 − 196. i)12-s − 791.·13-s + 2.39e3i·14-s + (633. + 207. i)15-s − 3.54e3·16-s + 1.19e3i·17-s + ⋯
L(s)  = 1  − 0.938i·2-s + (0.311 − 0.950i)3-s + 0.119·4-s + 0.197i·5-s + (−0.891 − 0.292i)6-s − 0.928·7-s − 1.05i·8-s + (−0.806 − 0.591i)9-s + 0.185·10-s − 0.534i·11-s + (0.0372 − 0.113i)12-s − 0.360·13-s + 0.871i·14-s + (0.187 + 0.0615i)15-s − 0.865·16-s + 0.242i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $-0.950 - 0.311i$
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: \((2,\ 51,\ (\ :3),\ -0.950 - 0.311i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.234966 + 1.47135i\)
\(L(\frac12)\) \(\approx\) \(0.234966 + 1.47135i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-8.40 + 25.6i)T \)
17 \( 1 - 1.19e3iT \)
good2 \( 1 + 7.50iT - 64T^{2} \)
5 \( 1 - 24.7iT - 1.56e4T^{2} \)
7 \( 1 + 318.T + 1.17e5T^{2} \)
11 \( 1 + 711. iT - 1.77e6T^{2} \)
13 \( 1 + 791.T + 4.82e6T^{2} \)
19 \( 1 + 2.30e3T + 4.70e7T^{2} \)
23 \( 1 + 1.95e3iT - 1.48e8T^{2} \)
29 \( 1 + 3.52e3iT - 5.94e8T^{2} \)
31 \( 1 + 1.30e3T + 8.87e8T^{2} \)
37 \( 1 - 2.61e4T + 2.56e9T^{2} \)
41 \( 1 + 6.89e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.19e5T + 6.32e9T^{2} \)
47 \( 1 + 1.88e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.62e4iT - 2.21e10T^{2} \)
59 \( 1 + 2.19e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.52e5T + 5.15e10T^{2} \)
67 \( 1 + 6.48e4T + 9.04e10T^{2} \)
71 \( 1 - 5.09e4iT - 1.28e11T^{2} \)
73 \( 1 + 2.37e5T + 1.51e11T^{2} \)
79 \( 1 + 6.29e5T + 2.43e11T^{2} \)
83 \( 1 + 2.50e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.90e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.40e6T + 8.32e11T^{2} \)
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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

−13.17142285351488315187385947221, −12.52920540050799482389508589851, −11.43934414208871575750945075324, −10.23415585285898871881825855814, −8.887503433000703601263302208965, −7.22097441794737533444680760204, −6.20043391820760406171616675412, −3.45844878198169989744758538994, −2.31866509873402669062228895573, −0.59869816264911509394768018310, 2.77132483397960888952196964188, 4.64600212318102123403481114487, 6.04957977493515452296330022407, 7.43403179873349448407987139136, 8.826762825915878064173307668382, 9.921384025846937142889544173635, 11.21544547282146075015019235931, 12.74708937717364890337481382037, 14.23593740497449026071408461298, 15.06897531735356295238230503639

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