↑

Properties

Label 1-23e2-529.208-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

Related objects

Downloads

Learn more

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: $\chi_{529} (208, \cdot )$
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 \( 1 + (0.682 + 0.730i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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