Properties

Label 1-23e2-529.351-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: $\chi_{529} (351, \cdot )$
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

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

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