Properties

Label 1-712-712.67-r1-0-0
Degree $1$
Conductor $712$
Sign $0.524 + 0.851i$
Analytic cond. $76.5150$
Root an. cond. $76.5150$
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  + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)5-s + (0.959 + 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.654 + 0.755i)13-s + (−0.415 − 0.909i)15-s + (0.415 − 0.909i)17-s + (−0.142 + 0.989i)19-s + (−0.415 − 0.909i)21-s + (0.142 − 0.989i)23-s + (0.841 + 0.540i)25-s + (0.841 − 0.540i)27-s + (0.959 + 0.281i)29-s + (0.142 + 0.989i)31-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)5-s + (0.959 + 0.281i)7-s + (−0.142 + 0.989i)9-s + (−0.959 + 0.281i)11-s + (0.654 + 0.755i)13-s + (−0.415 − 0.909i)15-s + (0.415 − 0.909i)17-s + (−0.142 + 0.989i)19-s + (−0.415 − 0.909i)21-s + (0.142 − 0.989i)23-s + (0.841 + 0.540i)25-s + (0.841 − 0.540i)27-s + (0.959 + 0.281i)29-s + (0.142 + 0.989i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.524 + 0.851i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ 0.524 + 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.705782718 + 0.9528310993i\)
\(L(\frac12)\) \(\approx\) \(1.705782718 + 0.9528310993i\)
\(L(1)\) \(\approx\) \(1.122464496 + 0.04632429883i\)
\(L(1)\) \(\approx\) \(1.122464496 + 0.04632429883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (-0.654 - 0.755i)T \)
5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.654 + 0.755i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.142 + 0.989i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.654 - 0.755i)T \)
53 \( 1 + (0.654 + 0.755i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.142 - 0.989i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−21.98766534515971580396244255190, −21.43752914056301081288857952576, −20.852735089939611488727462879623, −20.2368491110551331942731851092, −18.81657231696713474322894474551, −17.73123711390026179468817269051, −17.53215669845006645224238701072, −16.70525173406793116478262388342, −15.60216384695588222034829673862, −15.12414027390071213789247837932, −13.90205892684888899020457946111, −13.26649968057438554326706853097, −12.23872566125495054380480500704, −11.12220271225186520657727537547, −10.57571001988739883964427964588, −9.89186861525082756298458242679, −8.768141691170483506179088502163, −8.02213068856954907670003734383, −6.64730902412175689395915758488, −5.5373962854132210389512598817, −5.25401795195542146610840759653, −4.14770191142424768249350005130, −2.96060740298831301741495908538, −1.59673488295203595737920971815, −0.51303545371340131864609331799, 1.18371842115600220733997377702, 1.932839934578364202985157322546, 2.88072816023209604149617661008, 4.7339857044831415250665760537, 5.31168350812619945694933748927, 6.25739008773874602963733490849, 7.04490420448016872445735111524, 8.05691672419055103072427830597, 8.886260631343311117009642895612, 10.31548823064526861347932107197, 10.69681485690642646135561367380, 11.86280416060337265891481402131, 12.41452212260675529613613197708, 13.663772387703649625070019307221, 13.94331082144045856491128892077, 15.00188723173561819522570205325, 16.26103576247044347587226094801, 16.88423787655984474931442991688, 17.94575262187152242627243738903, 18.32429120616164018184182207735, 18.78209153811312091212242267288, 20.23007476471380923475778196609, 21.18812006660069434661292028416, 21.49458353813184734446262924384, 22.75314294811476063761365425487

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