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Properties

Label 1-712-712.405-r0-0-0
Degree $1$
Conductor $712$
Sign $0.155 + 0.987i$
Analytic cond. $3.30651$
Root an. cond. $3.30651$
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  + (−0.540 + 0.841i)3-s + (−0.654 + 0.755i)5-s + (0.755 + 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.540 − 0.841i)13-s + (−0.281 − 0.959i)15-s + (0.959 + 0.281i)17-s + (0.909 − 0.415i)19-s + (−0.959 + 0.281i)21-s + (0.909 − 0.415i)23-s + (−0.142 − 0.989i)25-s + (0.989 + 0.142i)27-s + (0.755 + 0.654i)29-s + (0.909 + 0.415i)31-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)3-s + (−0.654 + 0.755i)5-s + (0.755 + 0.654i)7-s + (−0.415 − 0.909i)9-s + (0.654 + 0.755i)11-s + (0.540 − 0.841i)13-s + (−0.281 − 0.959i)15-s + (0.959 + 0.281i)17-s + (0.909 − 0.415i)19-s + (−0.959 + 0.281i)21-s + (0.909 − 0.415i)23-s + (−0.142 − 0.989i)25-s + (0.989 + 0.142i)27-s + (0.755 + 0.654i)29-s + (0.909 + 0.415i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(3.30651\)
Root analytic conductor: \(3.30651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (405, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (0:\ ),\ 0.155 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.008501635 + 0.8620962561i\)
\(L(\frac12)\) \(\approx\) \(1.008501635 + 0.8620962561i\)
\(L(1)\) \(\approx\) \(0.9217281501 + 0.4267041724i\)
\(L(1)\) \(\approx\) \(0.9217281501 + 0.4267041724i\)

Euler product

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

−22.88268715121453236160936830990, −21.44098953359853355958644291553, −20.8347871845103624248691091146, −19.81902544903228645832230587250, −19.15891541753942710228053327031, −18.429553975646301160774276763817, −17.409268095019091582955943725, −16.627750074173798718910803750145, −16.329884876352565287624586072989, −14.94729759484967925292778345145, −13.756925121806687653191379964773, −13.56586335776521498120911154149, −12.18397513864253017035934281744, −11.63182824733549404190528770883, −11.16633214373115347740120238976, −9.82182115760188463144210748473, −8.58027218074438246582502267816, −7.97216417022190775811888128358, −7.15943932498384571701758538679, −6.15339091406996306058583460409, −5.133460774688861675962282559536, −4.30543267221756154379366005995, −3.16135311606256867414139593619, −1.335181082779856148389376513, −1.04086172827876752490743064263, 1.10134511520505392567010293192, 2.79551710462389808336093326618, 3.59752069672525045539819336728, 4.65943465797310130769465645668, 5.43855214531585219756609615932, 6.460744642224043345437630060574, 7.46589303092183936154852574441, 8.47835419759785757665911085365, 9.3859154474453058427127246662, 10.48158819874247604407636877696, 10.96458662796382954557218139170, 11.995286928001184203337004138, 12.33573316676060258649690299324, 14.08869129155272869373299734176, 14.756676113747179294831519851, 15.413313763899919208482138018459, 15.98644273975968658298701212325, 17.18722706278213875432437621240, 17.85246766436362957271257518383, 18.5310259647239898777074614093, 19.62199413672401864783394869092, 20.48258618887201655812012232615, 21.24438806546266612105305667328, 22.046259436026776518969811238229, 22.849572007015029469212853196462

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