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Properties

Label 1-23e2-529.147-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: $\chi_{529} (147, \cdot )$
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

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

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