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Properties

Label 1-763-763.174-r0-0-0
Degree $1$
Conductor $763$
Sign $0.817 + 0.576i$
Analytic cond. $3.54335$
Root an. cond. $3.54335$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.597 + 0.802i)3-s + (0.939 − 0.342i)4-s + (0.686 + 0.727i)5-s + (0.448 − 0.893i)6-s + (−0.866 + 0.5i)8-s + (−0.286 − 0.957i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (−0.286 + 0.957i)12-s + (0.230 + 0.973i)13-s + (−0.993 + 0.116i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + (0.448 + 0.893i)18-s + (0.342 − 0.939i)19-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.597 + 0.802i)3-s + (0.939 − 0.342i)4-s + (0.686 + 0.727i)5-s + (0.448 − 0.893i)6-s + (−0.866 + 0.5i)8-s + (−0.286 − 0.957i)9-s + (−0.802 − 0.597i)10-s + (0.802 − 0.597i)11-s + (−0.286 + 0.957i)12-s + (0.230 + 0.973i)13-s + (−0.993 + 0.116i)15-s + (0.766 − 0.642i)16-s + (−0.642 − 0.766i)17-s + (0.448 + 0.893i)18-s + (0.342 − 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(763\)    =    \(7 \cdot 109\)
Sign: $0.817 + 0.576i$
Analytic conductor: \(3.54335\)
Root analytic conductor: \(3.54335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{763} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 763,\ (0:\ ),\ 0.817 + 0.576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8407459810 + 0.2667735343i\)
\(L(\frac12)\) \(\approx\) \(0.8407459810 + 0.2667735343i\)
\(L(1)\) \(\approx\) \(0.6828018684 + 0.2103812405i\)
\(L(1)\) \(\approx\) \(0.6828018684 + 0.2103812405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (0.686 + 0.727i)T \)
11 \( 1 + (0.802 - 0.597i)T \)
13 \( 1 + (0.230 + 0.973i)T \)
17 \( 1 + (-0.642 - 0.766i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.993 + 0.116i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
37 \( 1 + (-0.727 - 0.686i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.998 + 0.0581i)T \)
53 \( 1 + (0.727 - 0.686i)T \)
59 \( 1 + (-0.116 - 0.993i)T \)
61 \( 1 + (-0.0581 - 0.998i)T \)
67 \( 1 + (0.230 - 0.973i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (-0.597 - 0.802i)T \)
79 \( 1 + (-0.957 + 0.286i)T \)
83 \( 1 + (0.396 + 0.918i)T \)
89 \( 1 + (-0.893 - 0.448i)T \)
97 \( 1 + (0.286 - 0.957i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.26999835042133380964699801197, −21.40703933543244980164977384487, −20.45136831520624834135768624761, −19.78844412422355890947669327496, −19.098999532988152603003938144944, −17.91804207780864797987796253103, −17.63056561832208218171042443873, −17.02707280887904656337100453782, −16.17089539179659199588811195700, −15.2522277377767977258188962376, −13.928410154519777736714162080337, −13.0332858449760765499630478839, −12.24904465561021224513585250052, −11.73590565409075242973911820489, −10.49495104897686564609897701779, −9.96348826077409657868748273417, −8.82279658220804202480685705362, −8.13077022813968515648845319625, −7.220342038654808725435680582265, −6.17775966725539756076212341600, −5.69389931097161988071854632341, −4.25769423174535622579745219430, −2.69211900471290516404584299707, −1.6077156490571340204263775366, −1.04741364546317530945942658455, 0.78012224511987943207434378124, 2.21962017512955722519438747347, 3.215642800673897550430447798579, 4.52896654612282660814968690047, 5.689715044074200184926955315163, 6.61407687419657730479317924200, 6.88715813554358859899792309675, 8.621701076471269818718371100111, 9.206165920336532359372586748284, 9.936920654084349513790307154860, 10.83252513398598405749230569784, 11.35290461701886004849450439496, 12.13427188571703725366486822963, 13.89291219234450094306936782335, 14.365094832242776384716724785703, 15.50748604888603534505459309866, 16.07324632828597024020540460139, 16.9375883711814808073767010637, 17.58036313568222760530059687065, 18.24367574749194673879101851089, 19.0909764618388247548454498700, 20.002377030243722538593844432566, 20.95692345618662307912078123915, 21.63980577736120349629595285239, 22.29133144648865090735529472009

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