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Properties

Label 1-23e2-529.169-r0-0-0
Degree $1$
Conductor $529$
Sign $0.953 + 0.300i$
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  + (0.323 + 0.946i)2-s + (0.995 + 0.0991i)3-s + (−0.791 + 0.611i)4-s + (0.00620 − 0.999i)5-s + (0.227 + 0.973i)6-s + (0.717 − 0.696i)7-s + (−0.834 − 0.551i)8-s + (0.980 + 0.197i)9-s + (0.948 − 0.317i)10-s + (0.346 + 0.938i)11-s + (−0.847 + 0.530i)12-s + (−0.381 − 0.924i)13-s + (0.890 + 0.454i)14-s + (0.105 − 0.994i)15-s + (0.251 − 0.967i)16-s + (0.179 − 0.983i)17-s + ⋯
L(s)  = 1  + (0.323 + 0.946i)2-s + (0.995 + 0.0991i)3-s + (−0.791 + 0.611i)4-s + (0.00620 − 0.999i)5-s + (0.227 + 0.973i)6-s + (0.717 − 0.696i)7-s + (−0.834 − 0.551i)8-s + (0.980 + 0.197i)9-s + (0.948 − 0.317i)10-s + (0.346 + 0.938i)11-s + (−0.847 + 0.530i)12-s + (−0.381 − 0.924i)13-s + (0.890 + 0.454i)14-s + (0.105 − 0.994i)15-s + (0.251 − 0.967i)16-s + (0.179 − 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.953 + 0.300i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.953 + 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.201592254 + 0.3390180048i\)
\(L(\frac12)\) \(\approx\) \(2.201592254 + 0.3390180048i\)
\(L(1)\) \(\approx\) \(1.617266533 + 0.3937179626i\)
\(L(1)\) \(\approx\) \(1.617266533 + 0.3937179626i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.323 + 0.946i)T \)
3 \( 1 + (0.995 + 0.0991i)T \)
5 \( 1 + (0.00620 - 0.999i)T \)
7 \( 1 + (0.717 - 0.696i)T \)
11 \( 1 + (0.346 + 0.938i)T \)
13 \( 1 + (-0.381 - 0.924i)T \)
17 \( 1 + (0.179 - 0.983i)T \)
19 \( 1 + (-0.0434 - 0.999i)T \)
29 \( 1 + (-0.596 - 0.802i)T \)
31 \( 1 + (-0.873 + 0.487i)T \)
37 \( 1 + (0.735 + 0.678i)T \)
41 \( 1 + (-0.263 - 0.964i)T \)
43 \( 1 + (-0.166 + 0.985i)T \)
47 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (0.948 + 0.317i)T \)
59 \( 1 + (0.503 + 0.863i)T \)
61 \( 1 + (0.751 + 0.659i)T \)
67 \( 1 + (-0.215 + 0.976i)T \)
71 \( 1 + (0.767 + 0.640i)T \)
73 \( 1 + (-0.596 + 0.802i)T \)
79 \( 1 + (-0.117 - 0.993i)T \)
83 \( 1 + (0.275 + 0.961i)T \)
89 \( 1 + (-0.972 - 0.233i)T \)
97 \( 1 + (0.524 - 0.851i)T \)
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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

−23.51908696707299030340455150489, −22.11054320615525591672881056920, −21.67755352007467522433216740688, −21.11883255229410303395860712277, −20.03096223672714053750860180110, −19.19275892762047115835912753543, −18.66466050686133345648224707995, −18.14076203497480514562837947494, −16.71702019800234114769983486024, −15.226310154732196260058864620771, −14.521561397381606286187205339694, −14.27196852050421158507344522892, −13.2066844467839201371282828426, −12.173541319694368311339550505469, −11.34767628157329125015781206509, −10.517326523283278049437667218442, −9.51125343679266001040505441949, −8.6987919824821867673437916061, −7.83834926534212379482270696983, −6.485618299867737581527189600949, −5.45462714339122525623012193495, −3.98396231760054439898879488069, −3.44015388962557648822838182943, −2.237762287328350280213676121371, −1.68690839087466032648104136697, 1.04657757682182499993699750417, 2.58797663112919569640927529703, 3.985990906038327063498353665421, 4.62664063572988381176074181848, 5.39209271866669933312554505607, 7.111551466741158493316188214885, 7.571988138658777913695881384374, 8.46121252459043486429166179333, 9.301876426066080947529726151636, 10.04117090224173408379713155098, 11.706112829088402988719876387361, 12.81996575589778187076142824618, 13.33171175867254604039204039400, 14.24188806413790058149155861329, 14.99174217033513322008236801296, 15.704532168164128739336658116752, 16.65994319665008504361997939150, 17.49507052773348964357616479294, 18.13926553863761399154381948821, 19.56948522225558221740208946310, 20.37833715706598667930107684886, 20.81375188634518111638317795075, 21.87189601309994224526107516417, 22.857122091792653608770470307695, 23.82948916299605929672670604773

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