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Properties

Label 1-23e2-529.211-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.777 + 0.628i$
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  + (0.901 − 0.432i)2-s + (−0.986 + 0.160i)3-s + (0.626 − 0.779i)4-s + (−0.982 − 0.185i)5-s + (−0.820 + 0.571i)6-s + (−0.449 + 0.893i)7-s + (0.227 − 0.973i)8-s + (0.948 − 0.317i)9-s + (−0.966 + 0.257i)10-s + (0.369 − 0.929i)11-s + (−0.492 + 0.870i)12-s + (−0.673 − 0.739i)13-s + (−0.0186 + 0.999i)14-s + (0.999 + 0.0248i)15-s + (−0.215 − 0.976i)16-s + (−0.635 + 0.771i)17-s + ⋯
L(s)  = 1  + (0.901 − 0.432i)2-s + (−0.986 + 0.160i)3-s + (0.626 − 0.779i)4-s + (−0.982 − 0.185i)5-s + (−0.820 + 0.571i)6-s + (−0.449 + 0.893i)7-s + (0.227 − 0.973i)8-s + (0.948 − 0.317i)9-s + (−0.966 + 0.257i)10-s + (0.369 − 0.929i)11-s + (−0.492 + 0.870i)12-s + (−0.673 − 0.739i)13-s + (−0.0186 + 0.999i)14-s + (0.999 + 0.0248i)15-s + (−0.215 − 0.976i)16-s + (−0.635 + 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.777 + 0.628i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.777 + 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01455211210 - 0.04115685191i\)
\(L(\frac12)\) \(\approx\) \(0.01455211210 - 0.04115685191i\)
\(L(1)\) \(\approx\) \(0.7658666332 - 0.2166591586i\)
\(L(1)\) \(\approx\) \(0.7658666332 - 0.2166591586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.901 - 0.432i)T \)
3 \( 1 + (-0.986 + 0.160i)T \)
5 \( 1 + (-0.982 - 0.185i)T \)
7 \( 1 + (-0.449 + 0.893i)T \)
11 \( 1 + (0.369 - 0.929i)T \)
13 \( 1 + (-0.673 - 0.739i)T \)
17 \( 1 + (-0.635 + 0.771i)T \)
19 \( 1 + (-0.263 + 0.964i)T \)
29 \( 1 + (-0.944 + 0.329i)T \)
31 \( 1 + (-0.907 - 0.421i)T \)
37 \( 1 + (-0.935 - 0.352i)T \)
41 \( 1 + (0.154 + 0.987i)T \)
43 \( 1 + (-0.311 + 0.950i)T \)
47 \( 1 + (-0.990 - 0.136i)T \)
53 \( 1 + (-0.966 - 0.257i)T \)
59 \( 1 + (0.992 - 0.123i)T \)
61 \( 1 + (-0.926 + 0.375i)T \)
67 \( 1 + (-0.972 - 0.233i)T \)
71 \( 1 + (-0.426 + 0.904i)T \)
73 \( 1 + (-0.944 - 0.329i)T \)
79 \( 1 + (0.922 - 0.386i)T \)
83 \( 1 + (0.503 + 0.863i)T \)
89 \( 1 + (0.700 + 0.713i)T \)
97 \( 1 + (0.645 + 0.763i)T \)
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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

−23.9458879794232686012327338208, −23.092982128785183343927690611489, −22.49459105428433701302962058761, −22.02067092131849245004361175337, −20.6804867557241924716503150059, −19.9096134643231965520945384795, −19.06981167626914504665634066876, −17.72610050648539503596477239557, −17.04948886156762826515606221352, −16.27708595875332365089930115185, −15.58465352631269065992165359255, −14.717033574978679104395705294079, −13.62777343739218782025609999009, −12.76361468741835501026883078431, −11.98741988161913468089085127378, −11.33303632145643939632295210800, −10.48621565259999902415793622151, −9.13031512632749030508010408346, −7.37675464415558718498239982930, −7.18957724827599769323328903763, −6.456145464576451094094347268021, −4.922890541508634834979433926, −4.45486133207840615959310599636, −3.51903750026126177665340617936, −1.98679240469604402697304168001, 0.01853328909067339266293767003, 1.6264544775543512735910048278, 3.18985786968589967835311515144, 3.938294103172050109818909686662, 5.031486290175344169814894717207, 5.83024685687613615613178268510, 6.57147670892189315363249014112, 7.81691356285812549471346145263, 9.13709229832115377635377139862, 10.27969022898366673166224853683, 11.1607226308707840641741415061, 11.75576668061333209477788284783, 12.63467512625345174262657822185, 13.00078835346146064941289210687, 14.73491832369641848940964871979, 15.168011316768694651832633539066, 16.184695458716571694437932583599, 16.61444298598947355213805479095, 18.067848614845932927361583604265, 19.05581050169705257890013220395, 19.518772689769859201281359138014, 20.635697555664760853939650259626, 21.618570983595653942903181209918, 22.25843735688642866197211533183, 22.7445948622708714501305632443

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