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Properties

Label 1-23e2-529.338-r0-0-0
Degree $1$
Conductor $529$
Sign $0.0299 + 0.999i$
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  + (0.00620 + 0.999i)2-s + (−0.535 + 0.844i)3-s + (−0.999 + 0.0124i)4-s + (0.606 + 0.794i)5-s + (−0.847 − 0.530i)6-s + (0.664 − 0.747i)7-s + (−0.0186 − 0.999i)8-s + (−0.426 − 0.904i)9-s + (−0.791 + 0.611i)10-s + (−0.514 − 0.857i)11-s + (0.524 − 0.851i)12-s + (0.227 − 0.973i)13-s + (0.751 + 0.659i)14-s + (−0.996 + 0.0868i)15-s + (0.999 − 0.0248i)16-s + (0.0558 + 0.998i)17-s + ⋯
L(s)  = 1  + (0.00620 + 0.999i)2-s + (−0.535 + 0.844i)3-s + (−0.999 + 0.0124i)4-s + (0.606 + 0.794i)5-s + (−0.847 − 0.530i)6-s + (0.664 − 0.747i)7-s + (−0.0186 − 0.999i)8-s + (−0.426 − 0.904i)9-s + (−0.791 + 0.611i)10-s + (−0.514 − 0.857i)11-s + (0.524 − 0.851i)12-s + (0.227 − 0.973i)13-s + (0.751 + 0.659i)14-s + (−0.996 + 0.0868i)15-s + (0.999 − 0.0248i)16-s + (0.0558 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.0299 + 0.999i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (338, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.0299 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8598707029 + 0.8344842863i\)
\(L(\frac12)\) \(\approx\) \(0.8598707029 + 0.8344842863i\)
\(L(1)\) \(\approx\) \(0.7855227806 + 0.5886100539i\)
\(L(1)\) \(\approx\) \(0.7855227806 + 0.5886100539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.00620 + 0.999i)T \)
3 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + (0.606 + 0.794i)T \)
7 \( 1 + (0.664 - 0.747i)T \)
11 \( 1 + (-0.514 - 0.857i)T \)
13 \( 1 + (0.227 - 0.973i)T \)
17 \( 1 + (0.0558 + 0.998i)T \)
19 \( 1 + (0.988 - 0.148i)T \)
29 \( 1 + (0.922 - 0.386i)T \)
31 \( 1 + (0.998 + 0.0496i)T \)
37 \( 1 + (-0.952 - 0.305i)T \)
41 \( 1 + (-0.239 - 0.970i)T \)
43 \( 1 + (0.948 - 0.317i)T \)
47 \( 1 + (0.460 + 0.887i)T \)
53 \( 1 + (-0.791 - 0.611i)T \)
59 \( 1 + (-0.907 - 0.421i)T \)
61 \( 1 + (0.975 - 0.221i)T \)
67 \( 1 + (0.735 + 0.678i)T \)
71 \( 1 + (-0.726 + 0.687i)T \)
73 \( 1 + (0.922 + 0.386i)T \)
79 \( 1 + (0.179 + 0.983i)T \)
83 \( 1 + (-0.873 + 0.487i)T \)
89 \( 1 + (-0.935 - 0.352i)T \)
97 \( 1 + (0.995 + 0.0991i)T \)
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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

−23.21959851921770386609539937243, −22.3633329036837784738178269897, −21.46043609094398203067517393155, −20.79240382450998562113159373342, −20.03719959548851685090290642959, −18.93228368580041274713920321651, −18.16240147804423335963510817410, −17.777306354385923433795422890097, −16.851454280716275427898987342384, −15.75567460258699274350681839358, −14.14518050160492289980174667947, −13.74043567933563642657972746391, −12.68976475200480326595710150708, −12.009392858395072740614914692640, −11.54965380082730872743292717387, −10.30464896084685312310663960298, −9.350869981452350901815627133041, −8.526103860054969843839219999, −7.54549935962717853957841865558, −6.14122556892169112576001190920, −5.00728318293887462716036590347, −4.75857811862170176271554132674, −2.736209641352539809321990124, −1.897340903181533829263310694077, −1.10158557549427255851985153443, 0.89416845680846597841034000582, 3.087620598608355447146730228753, 3.97564430172372630806373461819, 5.19498592392077415327972426948, 5.77114661321823934401976325372, 6.68544060368300305636830485216, 7.80623811432491794767597036253, 8.65970667634251533116856347252, 9.97390982598061586334810330507, 10.44403829458863671356942634436, 11.25589379500483099119990763505, 12.72756970880330939889762662805, 13.93988870575932041138976431361, 14.228140971268610931513225269433, 15.46963142483794759417736604388, 15.81295562861472690237718320199, 17.15715887797475704564678479295, 17.44018643058421610716353327585, 18.1986071164104547731063257742, 19.26277972878361240612855857536, 20.71397642503291678296461362098, 21.36667678853751194240435219848, 22.232208668719469087430811885959, 22.822075207755511281364268335702, 23.65557984913199234357584188922

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