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Properties

Label 1-71e2-5041.167-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.0362 + 0.999i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (0.871 + 0.491i)2-s + (−0.350 + 0.936i)3-s + (0.517 + 0.855i)4-s + (0.532 + 0.846i)5-s + (−0.765 + 0.643i)6-s + (0.803 − 0.594i)7-s + (0.0309 + 0.999i)8-s + (−0.753 − 0.656i)9-s + (0.0486 + 0.998i)10-s + (0.309 − 0.951i)11-s + (−0.982 + 0.184i)12-s + (0.871 − 0.491i)13-s + (0.992 − 0.123i)14-s + (−0.979 + 0.202i)15-s + (−0.463 + 0.885i)16-s + (−0.988 − 0.149i)17-s + ⋯
L(s)  = 1  + (0.871 + 0.491i)2-s + (−0.350 + 0.936i)3-s + (0.517 + 0.855i)4-s + (0.532 + 0.846i)5-s + (−0.765 + 0.643i)6-s + (0.803 − 0.594i)7-s + (0.0309 + 0.999i)8-s + (−0.753 − 0.656i)9-s + (0.0486 + 0.998i)10-s + (0.309 − 0.951i)11-s + (−0.982 + 0.184i)12-s + (0.871 − 0.491i)13-s + (0.992 − 0.123i)14-s + (−0.979 + 0.202i)15-s + (−0.463 + 0.885i)16-s + (−0.988 − 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0362 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.0362 + 0.999i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.0362 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.680008739 + 2.584685257i\)
\(L(\frac12)\) \(\approx\) \(2.680008739 + 2.584685257i\)
\(L(1)\) \(\approx\) \(1.637254857 + 1.097430888i\)
\(L(1)\) \(\approx\) \(1.637254857 + 1.097430888i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.871 + 0.491i)T \)
3 \( 1 + (-0.350 + 0.936i)T \)
5 \( 1 + (0.532 + 0.846i)T \)
7 \( 1 + (0.803 - 0.594i)T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.871 - 0.491i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (0.981 - 0.193i)T \)
23 \( 1 + (0.562 - 0.826i)T \)
29 \( 1 + (-0.753 + 0.656i)T \)
31 \( 1 + (-0.540 - 0.841i)T \)
37 \( 1 + (0.562 - 0.826i)T \)
41 \( 1 + (0.759 - 0.650i)T \)
43 \( 1 + (-0.249 - 0.968i)T \)
47 \( 1 + (0.987 - 0.158i)T \)
53 \( 1 + (-0.0574 + 0.998i)T \)
59 \( 1 + (-0.399 + 0.916i)T \)
61 \( 1 + (-0.0398 + 0.999i)T \)
67 \( 1 + (-0.317 - 0.948i)T \)
73 \( 1 + (0.997 + 0.0707i)T \)
79 \( 1 + (0.989 - 0.141i)T \)
83 \( 1 + (-0.705 - 0.708i)T \)
89 \( 1 + (0.673 + 0.739i)T \)
97 \( 1 + (0.154 + 0.988i)T \)
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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

−17.97500981182150306916565619468, −17.38543356695001048139094669369, −16.5538416463646800098375564239, −15.78375727223041227857186926513, −15.09015917950061751098622042601, −14.2325833478719785800324035479, −13.77589104668203032341235608776, −13.009244808285872460030068164524, −12.698633211308338833309828018271, −11.840157358005007205529901800438, −11.43407083988265499962723256156, −10.89350263255989991947429488366, −9.65463618578855918670848330578, −9.21049447954587150454818290502, −8.30515168548974412698486806471, −7.49815386658601978538387648725, −6.637975109288055070926805853636, −6.03972215551522130723599308757, −5.33064865089433536909779122290, −4.832674000267648473774149877397, −4.098989984571472136658102705432, −2.93243418426305934616073786924, −1.95215625367741220521826442854, −1.66737678537104629437881937274, −0.99847307858247881164973852337, 0.85145517660258550683515667560, 2.18998837650069250428439177993, 2.99675109084811188828803779236, 3.73174770538034165438139225974, 4.18741055359213387436701421285, 5.19473811296795916108950905458, 5.69478545223120824023820337219, 6.280908137461194279409249370407, 7.10302274760328125636164147960, 7.758683066283923407844154181169, 8.8348718295114039137731127408, 9.18753799546974590199433409059, 10.68328855749734268937032206756, 10.764588651238612432605963405136, 11.27694983538009592490875376569, 12.04434430665285673192122953795, 13.19800154528013221536959997733, 13.74124751368676171757187464221, 14.209220985731370088175973972505, 14.92916978633641987798688432377, 15.3629066115120555643012267051, 16.18866951729584345583528021211, 16.73012187230441271410500447575, 17.35024210095962458947462223305, 18.01201735409881185733159702302

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