Properties

Degree $2$
Conductor $51$
Sign $0.970 + 0.242i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s i·3-s − 4-s i·6-s + 4i·7-s − 3·8-s − 9-s − 4i·11-s + i·12-s + 2·13-s + 4i·14-s − 16-s + (1 − 4i)17-s − 18-s − 4·19-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s − 0.5·4-s − 0.408i·6-s + 1.51i·7-s − 1.06·8-s − 0.333·9-s − 1.20i·11-s + 0.288i·12-s + 0.554·13-s + 1.06i·14-s − 0.250·16-s + (0.242 − 0.970i)17-s − 0.235·18-s − 0.917·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51\)    =    \(3 \cdot 17\)
Sign: $0.970 + 0.242i$
Motivic weight: \(1\)
Character: $\chi_{51} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 51,\ (\ :1/2),\ 0.970 + 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.977239 - 0.120303i\)
\(L(\frac12)\) \(\approx\) \(0.977239 - 0.120303i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
17 \( 1 + (-1 + 4i)T \)
good2 \( 1 - T + 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−15.22421906455026663818725423192, −14.20335462092373922512270459800, −13.22313622696751046955945037970, −12.29589703833658958394204491211, −11.29197683349063617608596857547, −9.170445946119745654646751668784, −8.397074917737072518802833061281, −6.25244206026251981780374239300, −5.25757798355882877965671943400, −3.08123160264227815861676586344, 3.83132277647941273193838504179, 4.72410306496764483411551904974, 6.58665318563334711739225264874, 8.353938731587443077985698253799, 9.868263186905898811129796321835, 10.77052143340783570038150502125, 12.48071732359980117615625341463, 13.37882699443855431166316779060, 14.47208214981668008954963250491, 15.20751821432646299185542006981

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