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Properties

Label 1-53-53.29-r0-0-0
Degree $1$
Conductor $53$
Sign $0.661 - 0.750i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(53\)
Sign: $0.661 - 0.750i$
Analytic conductor: \(0.246130\)
Root analytic conductor: \(0.246130\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{53} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 53,\ (0:\ ),\ 0.661 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.236845320 - 0.5587380035i\)
\(L(\frac12)\) \(\approx\) \(1.236845320 - 0.5587380035i\)
\(L(1)\) \(\approx\) \(1.423441895 - 0.4790189552i\)
\(L(1)\) \(\approx\) \(1.423441895 - 0.4790189552i\)

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

−33.256471689477561421266750899225, −32.062774234776306939715002062307, −31.62078827692844405414303688836, −30.201601170939439078808912486705, −29.739366006987139767181936723365, −27.288798564010742510682226927302, −26.33096518745927997051199676041, −25.59870846340394504961146063067, −24.20657359384256771343642812133, −23.35402128438882355851238289504, −22.20238672518309289266955562331, −20.732315588746661965491068509864, −19.64919552095740117574543729207, −18.45074227930956082025822513153, −16.54048575186248307020848209044, −15.53059205600777796486764633243, −14.4811114855070342889282448007, −13.37122959979086453819722696065, −12.31237074805663320758196860852, −10.422758805601985713979860655, −8.30396435522982313869403269794, −7.57715314627455191216517734532, −6.15082024996564596899041872216, −3.84612458655730945550855453803, −3.15592854437488769555979078467, 2.28909532469213051880545905159, 3.70253302267545332677170907405, 4.90760994207449430459416675852, 7.073606494611830477535230243946, 8.90726409718599152241790213908, 9.941724889707072234876348454502, 11.77899516706372459708168542255, 12.74179249743772605802686340051, 13.99302666050391104059680543296, 15.37510692540990219046319073340, 15.963098229879985004654035338009, 18.65046419576632016805261121513, 19.50416672384921551576471480046, 20.408686400035355087132771845447, 21.445780915826564090403262804, 22.7254752130125627080240626475, 23.909355367601615026632560932116, 25.05798443776214705272611843430, 26.34632705560417923602033588853, 27.855503393503781296423305566129, 28.56714579750848741604779461210, 30.19556809390485931670495163140, 31.23306960153797071415209259436, 31.77047972495108479497762639040, 32.65166357303927987754294856195

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