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Properties

Label 1-763-763.2-r1-0-0
Degree $1$
Conductor $763$
Sign $-0.142 - 0.989i$
Analytic cond. $81.9957$
Root an. cond. $81.9957$
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  − i·2-s + (0.173 − 0.984i)3-s − 4-s + (0.173 − 0.984i)5-s + (−0.984 − 0.173i)6-s + i·8-s + (−0.939 − 0.342i)9-s + (−0.984 − 0.173i)10-s + (0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)15-s + 16-s + (−0.866 + 0.5i)17-s + (−0.342 + 0.939i)18-s + i·19-s + ⋯
L(s)  = 1  − i·2-s + (0.173 − 0.984i)3-s − 4-s + (0.173 − 0.984i)5-s + (−0.984 − 0.173i)6-s + i·8-s + (−0.939 − 0.342i)9-s + (−0.984 − 0.173i)10-s + (0.642 + 0.766i)11-s + (−0.173 + 0.984i)12-s + (−0.642 + 0.766i)13-s + (−0.939 − 0.342i)15-s + 16-s + (−0.866 + 0.5i)17-s + (−0.342 + 0.939i)18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 763 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.142 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(763\)    =    \(7 \cdot 109\)
Sign: $-0.142 - 0.989i$
Analytic conductor: \(81.9957\)
Root analytic conductor: \(81.9957\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{763} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 763,\ (1:\ ),\ -0.142 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.066135244 - 1.230588096i\)
\(L(\frac12)\) \(\approx\) \(1.066135244 - 1.230588096i\)
\(L(1)\) \(\approx\) \(0.7021368946 - 0.7134894970i\)
\(L(1)\) \(\approx\) \(0.7021368946 - 0.7134894970i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
109 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.642 + 0.766i)T \)
13 \( 1 + (-0.642 + 0.766i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + iT \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.939 - 0.342i)T \)
31 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (-0.342 + 0.939i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.642 - 0.766i)T \)
53 \( 1 + (0.342 + 0.939i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−22.500200579357093527857575417604, −21.83026931719158368784565095822, −21.21115033356144691308555070431, −19.69710535877970923323227037452, −19.358142828330311847571039465564, −17.92829110012915380239000822734, −17.62195653205634873489044774398, −16.55300235301946959118173768774, −15.837655324136990573401156120772, −15.09932828531210960180847874513, −14.443321609341704700265397016212, −13.878174547871305589936638028106, −12.84683402016190725717760328350, −11.36689099006555240666992665476, −10.69691147113809173427195902320, −9.74987858135721504944496938449, −9.02125439584767977809149670486, −8.213032389730347167639571479074, −7.033164698459898449700625291864, −6.38872901818019933702638634977, −5.294516148894262464993998855772, −4.57125663371033415184541105508, −3.39637266976974755733802686793, −2.7122707579173090746859153535, −0.51358884046369455585769169730, 0.75880124378688682409194701615, 1.73832938363195796569255335243, 2.26745221426963493886933866195, 3.75624508577416194016890989321, 4.6040436754341763660717671504, 5.62475663525388444266400504329, 6.76197962150547051184612052483, 7.85757904930324801192653604698, 8.78944945680796876048503308241, 9.31427449416566608170228763, 10.31056584945915965801589289837, 11.71304536986076031223181857688, 12.01192792128015844626793629552, 12.832706513411180004789442485671, 13.4995929299363072204624456485, 14.27181491820469022463947016288, 15.19934935941378423073716911152, 16.882693716955963060008706936499, 17.20816431304696376886881231621, 18.00864890980107104573221292025, 19.10314221787324160302371803254, 19.49612294226638264306634112427, 20.34848736757748510222360411421, 20.88238981384569171748213093489, 21.914110008004701608431956752611

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