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Properties

Label 1-23e2-529.18-r0-0-0
Degree $1$
Conductor $529$
Sign $0.641 + 0.767i$
Analytic cond. $2.45666$
Root an. cond. $2.45666$
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  + (−0.311 − 0.950i)2-s + (−0.514 + 0.857i)3-s + (−0.806 + 0.591i)4-s + (−0.966 + 0.257i)5-s + (0.975 + 0.221i)6-s + (0.606 + 0.794i)7-s + (0.813 + 0.581i)8-s + (−0.471 − 0.882i)9-s + (0.545 + 0.837i)10-s + (0.664 − 0.747i)11-s + (−0.0929 − 0.995i)12-s + (0.751 + 0.659i)13-s + (0.566 − 0.824i)14-s + (0.275 − 0.961i)15-s + (0.299 − 0.954i)16-s + (−0.287 + 0.957i)17-s + ⋯
L(s)  = 1  + (−0.311 − 0.950i)2-s + (−0.514 + 0.857i)3-s + (−0.806 + 0.591i)4-s + (−0.966 + 0.257i)5-s + (0.975 + 0.221i)6-s + (0.606 + 0.794i)7-s + (0.813 + 0.581i)8-s + (−0.471 − 0.882i)9-s + (0.545 + 0.837i)10-s + (0.664 − 0.747i)11-s + (−0.0929 − 0.995i)12-s + (0.751 + 0.659i)13-s + (0.566 − 0.824i)14-s + (0.275 − 0.961i)15-s + (0.299 − 0.954i)16-s + (−0.287 + 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.641 + 0.767i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.641 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6797002170 + 0.3177564209i\)
\(L(\frac12)\) \(\approx\) \(0.6797002170 + 0.3177564209i\)
\(L(1)\) \(\approx\) \(0.6920967511 + 0.03691540352i\)
\(L(1)\) \(\approx\) \(0.6920967511 + 0.03691540352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.311 - 0.950i)T \)
3 \( 1 + (-0.514 + 0.857i)T \)
5 \( 1 + (-0.966 + 0.257i)T \)
7 \( 1 + (0.606 + 0.794i)T \)
11 \( 1 + (0.664 - 0.747i)T \)
13 \( 1 + (0.751 + 0.659i)T \)
17 \( 1 + (-0.287 + 0.957i)T \)
19 \( 1 + (0.251 - 0.967i)T \)
29 \( 1 + (0.154 - 0.987i)T \)
31 \( 1 + (-0.820 + 0.571i)T \)
37 \( 1 + (0.992 + 0.123i)T \)
41 \( 1 + (-0.215 + 0.976i)T \)
43 \( 1 + (-0.726 + 0.687i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (0.545 - 0.837i)T \)
59 \( 1 + (0.984 + 0.172i)T \)
61 \( 1 + (0.392 + 0.919i)T \)
67 \( 1 + (0.955 + 0.293i)T \)
71 \( 1 + (-0.596 + 0.802i)T \)
73 \( 1 + (0.154 + 0.987i)T \)
79 \( 1 + (-0.239 + 0.970i)T \)
83 \( 1 + (-0.673 + 0.739i)T \)
89 \( 1 + (-0.885 + 0.465i)T \)
97 \( 1 + (0.346 - 0.938i)T \)
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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

−23.506849188314130440591410091408, −22.901114069804163215126403164042, −22.29466462173522932295528499327, −20.318431238059140785997923250867, −20.063379044238680783709129044066, −18.81402243847375976308932709162, −18.21369963604888639170814331275, −17.40660118637459496174776593315, −16.639913165370027618170625382851, −15.958240056106080969524901850884, −14.84795241124374606403713137800, −14.07059768104615572029235886849, −13.151173360109514341701906942676, −12.23098918457876920369324040809, −11.24788302581911200790231864856, −10.42090469541785344863472234894, −9.02728848842079488533824616317, −8.05107614300004073947567634330, −7.43798719073606040624174997910, −6.82526578634077974253535400028, −5.60359987755144212867029092806, −4.69866487471226670932473490384, −3.7349317313703723924149583022, −1.57662734205086250378910986559, −0.63141746206266288195493784471, 1.102019993648135296895669804239, 2.67772265295430640711310656226, 3.78828257391316211669665614409, 4.32254020472608733972446219224, 5.48487902270653865971590994555, 6.71191754788386348289248050704, 8.37638558173953101159389327410, 8.68489333311011849552320652688, 9.74776155054486039209889095244, 10.97258402855194966850696200545, 11.41152919432412512286012788071, 11.82683091122230565490550147156, 13.04022328487118146923465866802, 14.33518378169173464501967999104, 15.10688937981867034957644823667, 16.07839583164455496777358960866, 16.85481629042557689105800060809, 17.86713952983141123691619126803, 18.592496764636069146304090740782, 19.50461723303908093255628708763, 20.19393109464503679795437878727, 21.39270937516267313952157592255, 21.65476604732529035808627506224, 22.46504496875143755771329898609, 23.42079351561105710753057901984

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