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Properties

Label 1-712-712.237-r1-0-0
Degree $1$
Conductor $712$
Sign $-0.921 + 0.387i$
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  + (0.997 − 0.0713i)3-s + (−0.281 − 0.959i)5-s + (−0.877 − 0.479i)7-s + (0.989 − 0.142i)9-s + (−0.959 − 0.281i)11-s + (−0.0713 − 0.997i)13-s + (−0.349 − 0.936i)15-s + (0.909 − 0.415i)17-s + (−0.800 − 0.599i)19-s + (−0.909 − 0.415i)21-s + (0.599 − 0.800i)23-s + (−0.841 + 0.540i)25-s + (0.977 − 0.212i)27-s + (−0.479 + 0.877i)29-s + (0.599 + 0.800i)31-s + ⋯
L(s)  = 1  + (0.997 − 0.0713i)3-s + (−0.281 − 0.959i)5-s + (−0.877 − 0.479i)7-s + (0.989 − 0.142i)9-s + (−0.959 − 0.281i)11-s + (−0.0713 − 0.997i)13-s + (−0.349 − 0.936i)15-s + (0.909 − 0.415i)17-s + (−0.800 − 0.599i)19-s + (−0.909 − 0.415i)21-s + (0.599 − 0.800i)23-s + (−0.841 + 0.540i)25-s + (0.977 − 0.212i)27-s + (−0.479 + 0.877i)29-s + (0.599 + 0.800i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.921 + 0.387i$
Analytic conductor: \(76.5150\)
Root analytic conductor: \(76.5150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 712,\ (1:\ ),\ -0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2082238570 - 1.031831436i\)
\(L(\frac12)\) \(\approx\) \(-0.2082238570 - 1.031831436i\)
\(L(1)\) \(\approx\) \(0.9885607303 - 0.4694082001i\)
\(L(1)\) \(\approx\) \(0.9885607303 - 0.4694082001i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.997 - 0.0713i)T \)
5 \( 1 + (-0.281 - 0.959i)T \)
7 \( 1 + (-0.877 - 0.479i)T \)
11 \( 1 + (-0.959 - 0.281i)T \)
13 \( 1 + (-0.0713 - 0.997i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.800 - 0.599i)T \)
23 \( 1 + (0.599 - 0.800i)T \)
29 \( 1 + (-0.479 + 0.877i)T \)
31 \( 1 + (0.599 + 0.800i)T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-0.0713 + 0.997i)T \)
43 \( 1 + (-0.479 - 0.877i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (-0.997 - 0.0713i)T \)
61 \( 1 + (0.212 + 0.977i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.281 + 0.959i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (-0.989 - 0.142i)T \)
83 \( 1 + (-0.349 + 0.936i)T \)
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

−23.004745279552092902635653277392, −21.76833117461292508907196181870, −21.36125100040948871363337381415, −20.42965011014444003504851134743, −19.26844498678666318622998788089, −18.94843665974039459952387584583, −18.46819452859389840602744127531, −17.06514201946369001997224126178, −16.05257529525231677428208304100, −15.25760407230402195647059542829, −14.82174603986828038531003008846, −13.809896084803566675897861837097, −13.06067060260755329849034184769, −12.17845465436619476399626705870, −11.08894588682980482605290385152, −9.94363508645787624894728652935, −9.66465071964251172431374475723, −8.36454415746996783133942362968, −7.65152388772976633512657483880, −6.76370343123070029988345208997, −5.85635415063441623309788506693, −4.37512187636771690350393888133, −3.452445326268352093104806743903, −2.711110354011228697050843413546, −1.83295425665175944493591094306, 0.21014332557372108555093353283, 1.17452232834377087776677032314, 2.72586712010313663951179362857, 3.334985241499418066867109656036, 4.48556034390193259628414378845, 5.36968575283850349462910278881, 6.71116045160991590397142071862, 7.69990107305339481232866460382, 8.32170225978263440766758728315, 9.20133384010558285467215506897, 10.03944251653583953818982330286, 10.84266101149903124291421108310, 12.42930539349854482185738024114, 12.8964602324838182190487842348, 13.42913503200666213779153291142, 14.52998179067531011947256112464, 15.4570327474275635076325736750, 16.12240953067473363065186656290, 16.80056280362089625466816109975, 18.00040736209544333196366712205, 18.92936652387535944325209443328, 19.62701414779282880835317126201, 20.296222570523973789300619909911, 20.893726062114019916158932914999, 21.697202131922881329961773334560

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