Properties

Degree 1
Conductor 53
Sign $0.661 - 0.750i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(53\)
\( \varepsilon \)  =  $0.661 - 0.750i$
motivic weight  =  \(0\)
character  :  $\chi_{53} (29, \cdot )$
Sato-Tate  :  $\mu(26)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 53,\ (0:\ ),\ 0.661 - 0.750i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.236845320 - 0.5587380035i$
$L(\frac12,\chi)$  $\approx$  $1.236845320 - 0.5587380035i$
$L(\chi,1)$  $\approx$  1.423441895 - 0.4790189552i
$L(1,\chi)$  $\approx$  1.423441895 - 0.4790189552i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

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