Properties

Label 1-53-53.42-r0-0-0
Degree $1$
Conductor $53$
Sign $0.0198 - 0.999i$
Analytic cond. $0.246130$
Root an. cond. $0.246130$
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  + (−0.748 − 0.663i)2-s + (−0.970 + 0.239i)3-s + (0.120 + 0.992i)4-s + (0.885 − 0.464i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (−0.970 − 0.239i)10-s + (−0.354 − 0.935i)11-s + (−0.354 − 0.935i)12-s + (0.120 − 0.992i)13-s + (0.120 + 0.992i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (0.568 + 0.822i)17-s + ⋯
L(s)  = 1  + (−0.748 − 0.663i)2-s + (−0.970 + 0.239i)3-s + (0.120 + 0.992i)4-s + (0.885 − 0.464i)5-s + (0.885 + 0.464i)6-s + (−0.748 − 0.663i)7-s + (0.568 − 0.822i)8-s + (0.885 − 0.464i)9-s + (−0.970 − 0.239i)10-s + (−0.354 − 0.935i)11-s + (−0.354 − 0.935i)12-s + (0.120 − 0.992i)13-s + (0.120 + 0.992i)14-s + (−0.748 + 0.663i)15-s + (−0.970 + 0.239i)16-s + (0.568 + 0.822i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0198 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0198 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.0198 - 0.999i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.0198 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3424877843 - 0.3357424895i\)
\(L(\frac12)\) \(\approx\) \(0.3424877843 - 0.3357424895i\)
\(L(1)\) \(\approx\) \(0.5321343173 - 0.2590884114i\)
\(L(1)\) \(\approx\) \(0.5321343173 - 0.2590884114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad53 \( 1 \)
good2 \( 1 + (-0.748 - 0.663i)T \)
3 \( 1 + (-0.970 + 0.239i)T \)
5 \( 1 + (0.885 - 0.464i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
11 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (0.120 - 0.992i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (0.120 - 0.992i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.354 + 0.935i)T \)
31 \( 1 + (-0.354 + 0.935i)T \)
37 \( 1 + (-0.970 + 0.239i)T \)
41 \( 1 + (-0.354 - 0.935i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (0.885 + 0.464i)T \)
61 \( 1 + (0.568 - 0.822i)T \)
67 \( 1 + (0.120 + 0.992i)T \)
71 \( 1 + (-0.970 - 0.239i)T \)
73 \( 1 + (0.568 + 0.822i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + T \)
89 \( 1 + (0.568 + 0.822i)T \)
97 \( 1 + (0.885 - 0.464i)T \)
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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

−33.63653392228576428410457671312, −33.21517850730752107046545845792, −31.56265812826811872184449327221, −29.71350161177565455356170232231, −28.84166023274688506268830013847, −28.18959263650212335423838909959, −26.72909692290306220752907240828, −25.51887141735663028927245196427, −24.78744947082757948827579357058, −23.2431411018332705106758205792, −22.4860578960457206067325154375, −20.97635478765946152534987153602, −18.86796074542560018627741131963, −18.42162790634526835286821958221, −17.17421376683755996894774779983, −16.25419118492275221723013449483, −14.914272092351730503658025333235, −13.34563867407908958080626976786, −11.74270148953157072830785110419, −10.19787324118481789988163780681, −9.4126926796884784551721532653, −7.27598264495185618344386428422, −6.28920722496971948110984756166, −5.22957463881822154513079225271, −1.96871538591134264192016974721, 0.97746821334734765209104852133, 3.36204948659211704297753310604, 5.36210668094186783033092246027, 6.90382504700526516140081716810, 8.827293008149977338440979990240, 10.19234373902293297497398409098, 10.827353451752792306538451916761, 12.57046561809275041616870347469, 13.32766903011590739762540562570, 15.919698453633477940197285444362, 16.89528733251139442914043628834, 17.62700366054847927063015306288, 18.9315273523695577925989373120, 20.36287849153590311400708600391, 21.467684909945464116956206091716, 22.324170530821241466598758152757, 23.80821321931470503989917842514, 25.36329069258860242535166297818, 26.47738311554575424015530388813, 27.62428406831909709384136436779, 28.69987188637676019210713826759, 29.376487711791626276607628274333, 30.169610320196112711654389640973, 32.23197792765326368432791508397, 33.04342158238739243186610907298

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