Properties

Label 1-23e2-529.12-r0-0-0
Degree $1$
Conductor $529$
Sign $0.124 + 0.992i$
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  + (−0.896 + 0.443i)2-s + (0.566 + 0.824i)3-s + (0.606 − 0.794i)4-s + (0.437 − 0.899i)5-s + (−0.873 − 0.487i)6-s + (0.940 + 0.340i)7-s + (−0.191 + 0.981i)8-s + (−0.358 + 0.933i)9-s + (0.00620 + 0.999i)10-s + (0.645 + 0.763i)11-s + (0.998 + 0.0496i)12-s + (0.275 + 0.961i)13-s + (−0.993 + 0.111i)14-s + (0.988 − 0.148i)15-s + (−0.263 − 0.964i)16-s + (0.545 − 0.837i)17-s + ⋯
L(s)  = 1  + (−0.896 + 0.443i)2-s + (0.566 + 0.824i)3-s + (0.606 − 0.794i)4-s + (0.437 − 0.899i)5-s + (−0.873 − 0.487i)6-s + (0.940 + 0.340i)7-s + (−0.191 + 0.981i)8-s + (−0.358 + 0.933i)9-s + (0.00620 + 0.999i)10-s + (0.645 + 0.763i)11-s + (0.998 + 0.0496i)12-s + (0.275 + 0.961i)13-s + (−0.993 + 0.111i)14-s + (0.988 − 0.148i)15-s + (−0.263 − 0.964i)16-s + (0.545 − 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.124 + 0.992i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.124 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.024160159 + 0.9035379212i\)
\(L(\frac12)\) \(\approx\) \(1.024160159 + 0.9035379212i\)
\(L(1)\) \(\approx\) \(0.9557564272 + 0.4545600023i\)
\(L(1)\) \(\approx\) \(0.9557564272 + 0.4545600023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.896 + 0.443i)T \)
3 \( 1 + (0.566 + 0.824i)T \)
5 \( 1 + (0.437 - 0.899i)T \)
7 \( 1 + (0.940 + 0.340i)T \)
11 \( 1 + (0.645 + 0.763i)T \)
13 \( 1 + (0.275 + 0.961i)T \)
17 \( 1 + (0.545 - 0.837i)T \)
19 \( 1 + (0.0310 + 0.999i)T \)
29 \( 1 + (-0.426 + 0.904i)T \)
31 \( 1 + (-0.860 - 0.508i)T \)
37 \( 1 + (-0.556 - 0.831i)T \)
41 \( 1 + (-0.596 - 0.802i)T \)
43 \( 1 + (0.323 + 0.946i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (0.00620 - 0.999i)T \)
59 \( 1 + (0.735 + 0.678i)T \)
61 \( 1 + (-0.673 + 0.739i)T \)
67 \( 1 + (0.154 - 0.987i)T \)
71 \( 1 + (0.879 + 0.476i)T \)
73 \( 1 + (-0.426 - 0.904i)T \)
79 \( 1 + (-0.726 + 0.687i)T \)
83 \( 1 + (0.999 + 0.0248i)T \)
89 \( 1 + (0.0558 + 0.998i)T \)
97 \( 1 + (0.481 + 0.876i)T \)
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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

−23.48885701427957112882749945840, −22.1886496802420612400856973800, −21.44694764943108685858999729402, −20.53033155050187627614010408857, −19.82434042313829802806065252525, −18.906731077941764206250933856979, −18.42986017232878211204545256005, −17.33848770468660354068865284642, −17.26748745874307877727270098454, −15.50888988549863863984724880157, −14.69807605523035111743380026269, −13.80177735353657348883125811449, −13.0081525742737926382245106387, −11.83618965405288969738302358432, −11.07919889318645810447628136054, −10.33626855711344344123415916250, −9.16991053134892854596311198040, −8.307890044424058608637310911004, −7.603721634789967316529717903, −6.735636435531813544429769902012, −5.79470041913986238024035566201, −3.747748876825533343501176988641, −2.977371607644153942714948380913, −1.90723221726045575730250609681, −1.00643687306112773194049728174, 1.521924295844324745401894607555, 2.13842941917532415883045244877, 3.96004944122694005631306072438, 5.01690532837779047341655372748, 5.66879402759463895859946874928, 7.16810736433841969331090916772, 8.10923337090166939008831522035, 9.01896207324903862644037699272, 9.35561323205294275515413819594, 10.327048542723294572458807676521, 11.40259867487635076719100875476, 12.24921862741194519212521985129, 13.90848856932612154684297395630, 14.43456520056309657766586553383, 15.215135017645594494810527017672, 16.41150583725038722611246173082, 16.60734030908211781059018236421, 17.68021484200296413949731910938, 18.51779628970845376643608853317, 19.539027372815524574713402936914, 20.56332545833282710467192722575, 20.73757188149864977644862412893, 21.6881975977361253832435824699, 22.93367938915941343103159609247, 24.074028275062110130588155597351

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