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Properties

Label 1-23e2-529.139-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.678 - 0.734i$
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  + (0.682 − 0.730i)2-s + (0.682 − 0.730i)3-s + (−0.0682 − 0.997i)4-s + (−0.334 + 0.942i)5-s + (−0.0682 − 0.997i)6-s + (−0.0682 + 0.997i)7-s + (−0.775 − 0.631i)8-s + (−0.0682 − 0.997i)9-s + (0.460 + 0.887i)10-s + (−0.576 − 0.816i)11-s + (−0.775 − 0.631i)12-s + (0.460 − 0.887i)13-s + (0.682 + 0.730i)14-s + (0.460 + 0.887i)15-s + (−0.990 + 0.136i)16-s + (0.460 − 0.887i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (0.682 − 0.730i)3-s + (−0.0682 − 0.997i)4-s + (−0.334 + 0.942i)5-s + (−0.0682 − 0.997i)6-s + (−0.0682 + 0.997i)7-s + (−0.775 − 0.631i)8-s + (−0.0682 − 0.997i)9-s + (0.460 + 0.887i)10-s + (−0.576 − 0.816i)11-s + (−0.775 − 0.631i)12-s + (0.460 − 0.887i)13-s + (0.682 + 0.730i)14-s + (0.460 + 0.887i)15-s + (−0.990 + 0.136i)16-s + (0.460 − 0.887i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.678 - 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8316274991 - 1.899146182i\)
\(L(\frac12)\) \(\approx\) \(0.8316274991 - 1.899146182i\)
\(L(1)\) \(\approx\) \(1.248913752 - 1.032137984i\)
\(L(1)\) \(\approx\) \(1.248913752 - 1.032137984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.682 - 0.730i)T \)
3 \( 1 + (0.682 - 0.730i)T \)
5 \( 1 + (-0.334 + 0.942i)T \)
7 \( 1 + (-0.0682 + 0.997i)T \)
11 \( 1 + (-0.576 - 0.816i)T \)
13 \( 1 + (0.460 - 0.887i)T \)
17 \( 1 + (0.460 - 0.887i)T \)
19 \( 1 + (0.682 - 0.730i)T \)
29 \( 1 + (-0.576 - 0.816i)T \)
31 \( 1 + (0.962 + 0.269i)T \)
37 \( 1 + (-0.990 - 0.136i)T \)
41 \( 1 + (0.854 - 0.519i)T \)
43 \( 1 + (0.203 + 0.979i)T \)
47 \( 1 + (0.962 - 0.269i)T \)
53 \( 1 + (0.460 - 0.887i)T \)
59 \( 1 + (0.682 + 0.730i)T \)
61 \( 1 + (-0.334 + 0.942i)T \)
67 \( 1 + (-0.576 - 0.816i)T \)
71 \( 1 + (0.854 + 0.519i)T \)
73 \( 1 + (-0.576 + 0.816i)T \)
79 \( 1 + (0.203 + 0.979i)T \)
83 \( 1 + (-0.334 + 0.942i)T \)
89 \( 1 + (-0.917 - 0.398i)T \)
97 \( 1 + (0.854 + 0.519i)T \)
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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

−23.71406590196130153820051248538, −23.16434796630221336152410483218, −22.181908899432494134981518427577, −21.00723823894195884797306833500, −20.75683402222925750698119848910, −19.99209846838273729880794704368, −18.86716679908040926752026426413, −17.39415478371936379477032417076, −16.70632081388991131073695734907, −16.07411664920287327483288281431, −15.37501766197038058037897364437, −14.37550026645700862909193033962, −13.70104196627401997143467211023, −12.8932241693884785174260501114, −12.00054141733237519740521356263, −10.72741458484318464836331529749, −9.67554984709173400190900519644, −8.72520764753835145835480521333, −7.87386796132953484511758637786, −7.22855191564956595536496582451, −5.74583542529322433145379068593, −4.723709928791409520458612286043, −4.10733775886912813762859338681, −3.37132507185955420871843939519, −1.74869474386873057969602219343, 0.82284492167560614040994361004, 2.50393166186655155924911712594, 2.83382168468369996646855631000, 3.70016276315212744028638861615, 5.37755825087623655740627363122, 6.09172366882562917054103648280, 7.19079579102687538934547284197, 8.22788295916000118759741276886, 9.23706810364440404020488779255, 10.264622931067160584380997197026, 11.36374659893404039293529057797, 11.91705589421005770506964017974, 12.94051746007035215994326290113, 13.71214321469252652633517722649, 14.3738701212324947223923207166, 15.45066326664024235839303096858, 15.70221862558124954243152857323, 17.95994577203701272423700186921, 18.33319131695385432388937685943, 19.10662297274206611200837647187, 19.64784050070291463279912606871, 20.80122075599274347216390342284, 21.330776729865578812017574887907, 22.5693589753473418097940890318, 22.85854508137680940188214265230

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