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Properties

Label 1-23e2-529.323-r0-0-0
Degree $1$
Conductor $529$
Sign $0.945 + 0.326i$
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  + (−0.334 − 0.942i)2-s + (−0.334 − 0.942i)3-s + (−0.775 + 0.631i)4-s + (0.962 + 0.269i)5-s + (−0.775 + 0.631i)6-s + (−0.775 − 0.631i)7-s + (0.854 + 0.519i)8-s + (−0.775 + 0.631i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 − 0.136i)11-s + (0.854 + 0.519i)12-s + (−0.0682 + 0.997i)13-s + (−0.334 + 0.942i)14-s + (−0.0682 − 0.997i)15-s + (0.203 − 0.979i)16-s + (−0.0682 + 0.997i)17-s + ⋯
L(s)  = 1  + (−0.334 − 0.942i)2-s + (−0.334 − 0.942i)3-s + (−0.775 + 0.631i)4-s + (0.962 + 0.269i)5-s + (−0.775 + 0.631i)6-s + (−0.775 − 0.631i)7-s + (0.854 + 0.519i)8-s + (−0.775 + 0.631i)9-s + (−0.0682 − 0.997i)10-s + (−0.990 − 0.136i)11-s + (0.854 + 0.519i)12-s + (−0.0682 + 0.997i)13-s + (−0.334 + 0.942i)14-s + (−0.0682 − 0.997i)15-s + (0.203 − 0.979i)16-s + (−0.0682 + 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4528376341 + 0.07600175242i\)
\(L(\frac12)\) \(\approx\) \(0.4528376341 + 0.07600175242i\)
\(L(1)\) \(\approx\) \(0.5611720655 - 0.3004756372i\)
\(L(1)\) \(\approx\) \(0.5611720655 - 0.3004756372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.334 - 0.942i)T \)
3 \( 1 + (-0.334 - 0.942i)T \)
5 \( 1 + (0.962 + 0.269i)T \)
7 \( 1 + (-0.775 - 0.631i)T \)
11 \( 1 + (-0.990 - 0.136i)T \)
13 \( 1 + (-0.0682 + 0.997i)T \)
17 \( 1 + (-0.0682 + 0.997i)T \)
19 \( 1 + (-0.334 - 0.942i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (-0.917 + 0.398i)T \)
37 \( 1 + (0.203 + 0.979i)T \)
41 \( 1 + (0.682 + 0.730i)T \)
43 \( 1 + (0.460 + 0.887i)T \)
47 \( 1 + (-0.917 - 0.398i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (-0.334 + 0.942i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (-0.990 - 0.136i)T \)
71 \( 1 + (0.682 - 0.730i)T \)
73 \( 1 + (-0.990 + 0.136i)T \)
79 \( 1 + (0.460 + 0.887i)T \)
83 \( 1 + (0.962 + 0.269i)T \)
89 \( 1 + (-0.576 - 0.816i)T \)
97 \( 1 + (0.682 - 0.730i)T \)
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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

−23.318155605910650063081039555953, −22.54782765391488265540252163286, −22.07940786726889518733519819535, −20.95961490820827237144628182408, −20.289434594248339941517481650221, −18.9069273074631493848841801146, −18.069752331068172380732885344197, −17.48112345587708822272450670007, −16.3638051787521687865908881216, −16.06734140943989497895191890479, −15.10416663606074584035606786324, −14.361529600937127430666779020704, −13.17781109670591111573143578112, −12.547840616706678569625752374675, −10.87302990041533499685277346660, −10.074676865447442674844306480333, −9.48179552268942572097943270626, −8.75269732830102096269958163788, −7.57396165893002591671595671163, −6.25365844649514486575194455593, −5.507058570274457391658099885265, −5.13950296362499198723242544539, −3.652671407109027213151116022831, −2.32489953767535526757592390141, −0.28977184182540033709915824286, 1.30945116555659971782941984112, 2.252716102848426203515890337927, 3.11784399130489248323950573541, 4.55557145392512612330432180460, 5.79294345828472698310753929775, 6.75054059777871634863177888798, 7.64147254984706762254543248689, 8.81917321540986160900863202302, 9.71736620051219530830534337267, 10.683034269242096038801292214649, 11.20813511951261758279074385235, 12.52242400950805912407386697664, 13.21800103751057393394388886494, 13.52794027978246835920672784058, 14.63049590492687964717566009722, 16.43935031465405976173219493953, 16.99100397438909585385515440713, 17.8429516837117876733308662725, 18.521707445332682438354313802662, 19.27033887442998871734916639753, 19.94395720322929475357845534568, 21.06839007896123289991669296070, 21.794275524032632755798158249320, 22.54732739253226503805928349640, 23.48343888897266823967074221684

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