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Properties

Label 1-23e2-529.31-r0-0-0
Degree $1$
Conductor $529$
Sign $0.480 + 0.877i$
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  + (−0.847 − 0.530i)2-s + (−0.834 + 0.551i)3-s + (0.437 + 0.899i)4-s + (0.524 − 0.851i)5-s + (0.999 − 0.0248i)6-s + (0.827 + 0.561i)7-s + (0.105 − 0.994i)8-s + (0.392 − 0.919i)9-s + (−0.896 + 0.443i)10-s + (0.0806 + 0.996i)11-s + (−0.860 − 0.508i)12-s + (0.251 + 0.967i)13-s + (−0.404 − 0.914i)14-s + (0.0310 + 0.999i)15-s + (−0.616 + 0.787i)16-s + (−0.311 + 0.950i)17-s + ⋯
L(s)  = 1  + (−0.847 − 0.530i)2-s + (−0.834 + 0.551i)3-s + (0.437 + 0.899i)4-s + (0.524 − 0.851i)5-s + (0.999 − 0.0248i)6-s + (0.827 + 0.561i)7-s + (0.105 − 0.994i)8-s + (0.392 − 0.919i)9-s + (−0.896 + 0.443i)10-s + (0.0806 + 0.996i)11-s + (−0.860 − 0.508i)12-s + (0.251 + 0.967i)13-s + (−0.404 − 0.914i)14-s + (0.0310 + 0.999i)15-s + (−0.616 + 0.787i)16-s + (−0.311 + 0.950i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.480 + 0.877i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.480 + 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6453348556 + 0.3824828058i\)
\(L(\frac12)\) \(\approx\) \(0.6453348556 + 0.3824828058i\)
\(L(1)\) \(\approx\) \(0.6708103603 + 0.07569944593i\)
\(L(1)\) \(\approx\) \(0.6708103603 + 0.07569944593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.847 - 0.530i)T \)
3 \( 1 + (-0.834 + 0.551i)T \)
5 \( 1 + (0.524 - 0.851i)T \)
7 \( 1 + (0.827 + 0.561i)T \)
11 \( 1 + (0.0806 + 0.996i)T \)
13 \( 1 + (0.251 + 0.967i)T \)
17 \( 1 + (-0.311 + 0.950i)T \)
19 \( 1 + (0.664 + 0.747i)T \)
29 \( 1 + (-0.358 - 0.933i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (-0.944 - 0.329i)T \)
41 \( 1 + (0.980 - 0.197i)T \)
43 \( 1 + (-0.709 - 0.704i)T \)
47 \( 1 + (-0.576 + 0.816i)T \)
53 \( 1 + (-0.896 - 0.443i)T \)
59 \( 1 + (0.154 - 0.987i)T \)
61 \( 1 + (0.299 + 0.954i)T \)
67 \( 1 + (-0.471 + 0.882i)T \)
71 \( 1 + (0.586 - 0.809i)T \)
73 \( 1 + (-0.358 + 0.933i)T \)
79 \( 1 + (0.879 + 0.476i)T \)
83 \( 1 + (-0.263 + 0.964i)T \)
89 \( 1 + (0.545 - 0.837i)T \)
97 \( 1 + (-0.885 - 0.465i)T \)
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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

−23.46750393098399505172324144779, −22.688790709729178999641455686455, −21.87501211018465286101553728140, −20.68483159375174610625693731021, −19.73831464234334509389440515585, −18.6978842703602233511801825394, −18.082037356194009944180096088032, −17.656234819165479694555158480811, −16.78127158057394122020491177682, −15.95066573035404361148406135920, −14.899515124256244559631606724655, −13.90002841449287739492413549238, −13.35846867619638110318229468899, −11.61639885798433005860152231926, −11.03804386783144946494205245968, −10.52134203009597564250921804788, −9.39801484301538232777185242479, −8.11723985145352885559791138079, −7.37425762797921128764224943080, −6.63083609591894995454068955479, −5.68261258927416175517930960203, −4.986794011058456947509421175801, −2.99779456135551363217933468704, −1.713055567813017729592794345233, −0.63492412806240549957950101420, 1.39193790337194907900130464425, 1.992416133175955659883950512142, 3.86730774638212191744828341688, 4.68472392156237293634394718671, 5.730460056159189337102872190170, 6.80588055242483448546645466770, 8.093657803853589846482075374843, 9.02613140366017320655005191436, 9.65827640588145622443571674170, 10.53825550587691215971907058082, 11.52714305287314355209005253200, 12.17328454288948133037242312190, 12.835313756130388325544134769372, 14.3047378216839088899789501371, 15.49593296688693618737155205926, 16.230721207750193697974634195406, 17.13314819869789950073802992122, 17.62011923110218358642654992633, 18.291470661809088617526957004040, 19.40409135326332062759420069940, 20.64793863758042294061198586557, 20.93763683052876134769929287710, 21.67222633181225871398794884462, 22.49104921192739437998351778490, 23.744148118745220082192373478067

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