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Properties

Label 1-23e2-529.116-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.999 + 0.0178i$
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  + (−0.775 + 0.631i)2-s + (−0.775 + 0.631i)3-s + (0.203 − 0.979i)4-s + (0.854 + 0.519i)5-s + (0.203 − 0.979i)6-s + (0.203 + 0.979i)7-s + (0.460 + 0.887i)8-s + (0.203 − 0.979i)9-s + (−0.990 + 0.136i)10-s + (0.962 + 0.269i)11-s + (0.460 + 0.887i)12-s + (−0.990 − 0.136i)13-s + (−0.775 − 0.631i)14-s + (−0.990 + 0.136i)15-s + (−0.917 − 0.398i)16-s + (−0.990 − 0.136i)17-s + ⋯
L(s)  = 1  + (−0.775 + 0.631i)2-s + (−0.775 + 0.631i)3-s + (0.203 − 0.979i)4-s + (0.854 + 0.519i)5-s + (0.203 − 0.979i)6-s + (0.203 + 0.979i)7-s + (0.460 + 0.887i)8-s + (0.203 − 0.979i)9-s + (−0.990 + 0.136i)10-s + (0.962 + 0.269i)11-s + (0.460 + 0.887i)12-s + (−0.990 − 0.136i)13-s + (−0.775 − 0.631i)14-s + (−0.990 + 0.136i)15-s + (−0.917 − 0.398i)16-s + (−0.990 − 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.999 + 0.0178i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.999 + 0.0178i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.005727456793 + 0.6429317513i\)
\(L(\frac12)\) \(\approx\) \(0.005727456793 + 0.6429317513i\)
\(L(1)\) \(\approx\) \(0.4690956667 + 0.4333814784i\)
\(L(1)\) \(\approx\) \(0.4690956667 + 0.4333814784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.775 + 0.631i)T \)
3 \( 1 + (-0.775 + 0.631i)T \)
5 \( 1 + (0.854 + 0.519i)T \)
7 \( 1 + (0.203 + 0.979i)T \)
11 \( 1 + (0.962 + 0.269i)T \)
13 \( 1 + (-0.990 - 0.136i)T \)
17 \( 1 + (-0.990 - 0.136i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
29 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (0.682 - 0.730i)T \)
37 \( 1 + (-0.917 + 0.398i)T \)
41 \( 1 + (-0.0682 + 0.997i)T \)
43 \( 1 + (-0.576 + 0.816i)T \)
47 \( 1 + (0.682 + 0.730i)T \)
53 \( 1 + (-0.990 - 0.136i)T \)
59 \( 1 + (-0.775 - 0.631i)T \)
61 \( 1 + (0.854 + 0.519i)T \)
67 \( 1 + (0.962 + 0.269i)T \)
71 \( 1 + (-0.0682 - 0.997i)T \)
73 \( 1 + (0.962 - 0.269i)T \)
79 \( 1 + (-0.576 + 0.816i)T \)
83 \( 1 + (0.854 + 0.519i)T \)
89 \( 1 + (-0.334 + 0.942i)T \)
97 \( 1 + (-0.0682 - 0.997i)T \)
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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

−22.9600091488751908920610180439, −21.91550969705300965058723756242, −21.566051460179124774629815556814, −20.292380831339472927214099284, −19.66307247600072115600923934497, −18.93313415844127023595594344632, −17.53112101765794466896495117493, −17.431254288232365282734536401263, −16.89485793708082437193679105275, −15.85095171621752383863946586791, −14.07247281367401407517620001592, −13.45129534479594920568469953241, −12.52024439384514264644987355767, −11.833080201194038633550750572836, −10.78120304080219637299475461378, −10.218995057729422554149294202590, −9.105319324057063618151207530929, −8.2587605921114226759193804277, −6.92462165799894543725820396665, −6.60527056865706440125254904731, −4.9970077833155917468817818723, −4.13529640087704708723557545693, −2.390856990736522608003617637148, −1.55035227341893251881842221120, −0.4983028217385308085122671906, 1.5470850127084180222175567479, 2.64573493331438631218683900235, 4.52298700300367274954179713825, 5.33085224860207439176467011049, 6.39758337531976160624349350253, 6.65164252423378008705741385519, 8.24943238527486910305949778755, 9.34451753750536985668190767990, 9.74328770695239031923898798552, 10.71444459192700996605300182502, 11.55505023915553962121907548302, 12.50668887482924663927489257212, 14.069486893469197014212358112040, 14.89256630408580737025829273399, 15.336992874816840904459669001780, 16.45946596930055950181030865541, 17.37576473017661169072895797287, 17.62928261813872421565036241459, 18.57597926258437063500717910946, 19.44239032523908543469015102090, 20.60195980236466732343882061038, 21.61392673211256776548432059882, 22.2620210323087146311392146572, 22.90679474854566059338621989775, 24.141002300036613100457342751087

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