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Properties

Label 1-23e2-529.358-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.850 + 0.525i$
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  + (0.735 − 0.678i)2-s + (0.251 − 0.967i)3-s + (0.0806 − 0.996i)4-s + (−0.952 − 0.305i)5-s + (−0.471 − 0.882i)6-s + (0.700 − 0.713i)7-s + (−0.616 − 0.787i)8-s + (−0.873 − 0.487i)9-s + (−0.907 + 0.421i)10-s + (−0.404 − 0.914i)11-s + (−0.944 − 0.329i)12-s + (−0.759 + 0.650i)13-s + (0.0310 − 0.999i)14-s + (−0.535 + 0.844i)15-s + (−0.986 − 0.160i)16-s + (0.912 − 0.409i)17-s + ⋯
L(s)  = 1  + (0.735 − 0.678i)2-s + (0.251 − 0.967i)3-s + (0.0806 − 0.996i)4-s + (−0.952 − 0.305i)5-s + (−0.471 − 0.882i)6-s + (0.700 − 0.713i)7-s + (−0.616 − 0.787i)8-s + (−0.873 − 0.487i)9-s + (−0.907 + 0.421i)10-s + (−0.404 − 0.914i)11-s + (−0.944 − 0.329i)12-s + (−0.759 + 0.650i)13-s + (0.0310 − 0.999i)14-s + (−0.535 + 0.844i)15-s + (−0.986 − 0.160i)16-s + (0.912 − 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4333248978 - 1.525924348i\)
\(L(\frac12)\) \(\approx\) \(-0.4333248978 - 1.525924348i\)
\(L(1)\) \(\approx\) \(0.6851695624 - 1.159758945i\)
\(L(1)\) \(\approx\) \(0.6851695624 - 1.159758945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (0.735 - 0.678i)T \)
3 \( 1 + (0.251 - 0.967i)T \)
5 \( 1 + (-0.952 - 0.305i)T \)
7 \( 1 + (0.700 - 0.713i)T \)
11 \( 1 + (-0.404 - 0.914i)T \)
13 \( 1 + (-0.759 + 0.650i)T \)
17 \( 1 + (0.912 - 0.409i)T \)
19 \( 1 + (0.566 + 0.824i)T \)
29 \( 1 + (-0.847 + 0.530i)T \)
31 \( 1 + (0.948 - 0.317i)T \)
37 \( 1 + (0.901 - 0.432i)T \)
41 \( 1 + (-0.709 + 0.704i)T \)
43 \( 1 + (0.503 - 0.863i)T \)
47 \( 1 + (0.682 - 0.730i)T \)
53 \( 1 + (-0.907 - 0.421i)T \)
59 \( 1 + (-0.311 + 0.950i)T \)
61 \( 1 + (-0.117 - 0.993i)T \)
67 \( 1 + (0.130 + 0.991i)T \)
71 \( 1 + (-0.977 - 0.209i)T \)
73 \( 1 + (-0.847 - 0.530i)T \)
79 \( 1 + (-0.926 - 0.375i)T \)
83 \( 1 + (-0.166 + 0.985i)T \)
89 \( 1 + (0.717 - 0.696i)T \)
97 \( 1 + (0.798 - 0.601i)T \)
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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

−23.892964695750939251554864642504, −23.02364242607561475721481805079, −22.342826877209855731992095220305, −21.68763409478986637005913931507, −20.71296697598535207497308717919, −20.17141152904960308052118649166, −18.98006764244632609645366345019, −17.76751351598896395800673912337, −17.055797435953685822663069861192, −15.88658587377539644354466486757, −15.343832347616530152546885659715, −14.87517807645607754384980461800, −14.16321918766592949731134639027, −12.776705104221583780869884320335, −11.94251690546183685034527349061, −11.229428348089586675501416504790, −10.06554171668188027007551580176, −8.92090106016937921518249875615, −7.86231050153078236532898312423, −7.52833196612550110999817143565, −5.9278962696135011686207038207, −4.92899392323520756231381937725, −4.46610224273336545759852166053, −3.21724676743108845574232058297, −2.515883260525853758089242830552, 0.645025563967317804466581282077, 1.61774006626836546931513303701, 2.94711458210491233936800044060, 3.79454547657809840260273039510, 4.89390740039328637618492146814, 5.86210536558651418354827979158, 7.210736137437218921233663720999, 7.78640543270796793131175203531, 8.8697763973882998506075675748, 10.15117864732649281332396895194, 11.35748049071503444230515525327, 11.745642439526389309422821305687, 12.589042179260283261628362580502, 13.54464356367000038529629314864, 14.251125015686059573547066531893, 14.86384729656105887955036668094, 16.18861856352508450869360074776, 17.01487714224174051726765221807, 18.47090494030614208504490862446, 18.86574958431315680898351126358, 19.73393816867162376324669190975, 20.45229010017164178776429857253, 21.05058101561866105227812367635, 22.27027523970175789258335151354, 23.30458869238935652074675258481

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