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Properties

Label 1-23e2-529.101-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.319 + 0.947i$
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  + (0.867 + 0.498i)2-s + (−0.944 + 0.329i)3-s + (0.503 + 0.863i)4-s + (−0.215 − 0.976i)5-s + (−0.982 − 0.185i)6-s + (−0.239 + 0.970i)7-s + (0.00620 + 0.999i)8-s + (0.783 − 0.621i)9-s + (0.299 − 0.954i)10-s + (0.179 + 0.983i)11-s + (−0.759 − 0.650i)12-s + (0.901 − 0.432i)13-s + (−0.691 + 0.722i)14-s + (0.524 + 0.851i)15-s + (−0.492 + 0.870i)16-s + (−0.0186 − 0.999i)17-s + ⋯
L(s)  = 1  + (0.867 + 0.498i)2-s + (−0.944 + 0.329i)3-s + (0.503 + 0.863i)4-s + (−0.215 − 0.976i)5-s + (−0.982 − 0.185i)6-s + (−0.239 + 0.970i)7-s + (0.00620 + 0.999i)8-s + (0.783 − 0.621i)9-s + (0.299 − 0.954i)10-s + (0.179 + 0.983i)11-s + (−0.759 − 0.650i)12-s + (0.901 − 0.432i)13-s + (−0.691 + 0.722i)14-s + (0.524 + 0.851i)15-s + (−0.492 + 0.870i)16-s + (−0.0186 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.319 + 0.947i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.319 + 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9095895230 + 1.266695838i\)
\(L(\frac12)\) \(\approx\) \(0.9095895230 + 1.266695838i\)
\(L(1)\) \(\approx\) \(1.109784046 + 0.6296468272i\)
\(L(1)\) \(\approx\) \(1.109784046 + 0.6296468272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.867 + 0.498i)T \)
3 \( 1 + (-0.944 + 0.329i)T \)
5 \( 1 + (-0.215 - 0.976i)T \)
7 \( 1 + (-0.239 + 0.970i)T \)
11 \( 1 + (0.179 + 0.983i)T \)
13 \( 1 + (0.901 - 0.432i)T \)
17 \( 1 + (-0.0186 - 0.999i)T \)
19 \( 1 + (0.998 - 0.0496i)T \)
29 \( 1 + (-0.381 + 0.924i)T \)
31 \( 1 + (-0.514 + 0.857i)T \)
37 \( 1 + (0.586 - 0.809i)T \)
41 \( 1 + (0.0806 + 0.996i)T \)
43 \( 1 + (-0.404 + 0.914i)T \)
47 \( 1 + (-0.775 + 0.631i)T \)
53 \( 1 + (0.299 + 0.954i)T \)
59 \( 1 + (0.369 + 0.929i)T \)
61 \( 1 + (0.997 - 0.0744i)T \)
67 \( 1 + (0.969 + 0.245i)T \)
71 \( 1 + (0.700 + 0.713i)T \)
73 \( 1 + (-0.381 - 0.924i)T \)
79 \( 1 + (-0.834 + 0.551i)T \)
83 \( 1 + (0.346 - 0.938i)T \)
89 \( 1 + (0.392 + 0.919i)T \)
97 \( 1 + (-0.471 - 0.882i)T \)
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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

−23.14320929563520726459826731569, −22.4258658591366165531445582208, −21.86248761388239374373031737352, −20.93135890942270000734624273981, −19.83859576404351460658998781982, −18.92276807598283049141133056171, −18.54891006830902916332823359612, −17.254032259289784715842502929603, −16.33368721932971688981513549034, −15.59386115495539526730460023149, −14.4323943591670145719703292867, −13.58157939759085125244308938768, −13.1099499173327552056943453042, −11.701849915082588340000050450342, −11.2962023687417784783701058717, −10.57191301146287082895471693168, −9.810517104450483177736204957332, −7.93053817673918102123770689785, −6.83793080529251756216231388815, −6.29927468619407561110112245280, −5.44146520279297021465906360060, −3.94735431313048816817147806918, −3.57016365213652871044239909763, −1.99160815533901688135731479925, −0.76148143104779113463157278321, 1.43530800700077279595801778797, 3.073671290169894826680847785880, 4.21692780783849161968197025005, 5.13728468564875869114871965658, 5.572697969466209005215420084738, 6.65981132108115869954025440521, 7.65342163694467976706848182103, 8.88300498038686604589586868400, 9.67135327310649617988004433259, 11.21073805357216825555863812836, 11.83883841697384142977199208174, 12.6199486491817162317706152819, 13.11335832140589228445723725462, 14.5284377553584970930131791055, 15.50217250826230907365358055727, 16.07174098329616188256326233576, 16.5283825578756049093510684017, 17.8278612874324649955155466435, 18.16609782053079795230274055733, 19.95275038307524076455934023725, 20.61693325543432423637388261479, 21.42718273461682174088039554624, 22.22117135620639637856205397706, 23.00325792815864793140395671821, 23.465721327750069721067619967213

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