Properties

Label 1-23e2-529.377-r0-0-0
Degree $1$
Conductor $529$
Sign $-0.775 + 0.631i$
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  + (−0.759 + 0.650i)2-s + (0.00620 − 0.999i)3-s + (0.154 − 0.987i)4-s + (−0.471 − 0.882i)5-s + (0.645 + 0.763i)6-s + (−0.426 − 0.904i)7-s + (0.524 + 0.851i)8-s + (−0.999 − 0.0124i)9-s + (0.931 + 0.363i)10-s + (−0.311 − 0.950i)11-s + (−0.986 − 0.160i)12-s + (−0.492 − 0.870i)13-s + (0.912 + 0.409i)14-s + (−0.885 + 0.465i)15-s + (−0.952 − 0.305i)16-s + (−0.996 + 0.0868i)17-s + ⋯
L(s)  = 1  + (−0.759 + 0.650i)2-s + (0.00620 − 0.999i)3-s + (0.154 − 0.987i)4-s + (−0.471 − 0.882i)5-s + (0.645 + 0.763i)6-s + (−0.426 − 0.904i)7-s + (0.524 + 0.851i)8-s + (−0.999 − 0.0124i)9-s + (0.931 + 0.363i)10-s + (−0.311 − 0.950i)11-s + (−0.986 − 0.160i)12-s + (−0.492 − 0.870i)13-s + (0.912 + 0.409i)14-s + (−0.885 + 0.465i)15-s + (−0.952 − 0.305i)16-s + (−0.996 + 0.0868i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $-0.775 + 0.631i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ -0.775 + 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1102173061 - 0.3095919740i\)
\(L(\frac12)\) \(\approx\) \(-0.1102173061 - 0.3095919740i\)
\(L(1)\) \(\approx\) \(0.4367554419 - 0.2702240215i\)
\(L(1)\) \(\approx\) \(0.4367554419 - 0.2702240215i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.759 + 0.650i)T \)
3 \( 1 + (0.00620 - 0.999i)T \)
5 \( 1 + (-0.471 - 0.882i)T \)
7 \( 1 + (-0.426 - 0.904i)T \)
11 \( 1 + (-0.311 - 0.950i)T \)
13 \( 1 + (-0.492 - 0.870i)T \)
17 \( 1 + (-0.996 + 0.0868i)T \)
19 \( 1 + (-0.287 + 0.957i)T \)
29 \( 1 + (0.251 - 0.967i)T \)
31 \( 1 + (0.813 - 0.581i)T \)
37 \( 1 + (-0.673 - 0.739i)T \)
41 \( 1 + (0.783 + 0.621i)T \)
43 \( 1 + (0.626 + 0.779i)T \)
47 \( 1 + (0.460 - 0.887i)T \)
53 \( 1 + (0.931 - 0.363i)T \)
59 \( 1 + (0.751 - 0.659i)T \)
61 \( 1 + (0.940 - 0.340i)T \)
67 \( 1 + (-0.993 - 0.111i)T \)
71 \( 1 + (-0.0434 + 0.999i)T \)
73 \( 1 + (0.251 + 0.967i)T \)
79 \( 1 + (0.105 + 0.994i)T \)
83 \( 1 + (0.0806 - 0.996i)T \)
89 \( 1 + (-0.977 - 0.209i)T \)
97 \( 1 + (0.323 - 0.946i)T \)
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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

−23.92086695032090977509585273077, −22.57067217520517226334851232168, −22.23885424306874283052681359670, −21.49909503013941248572175920414, −20.63697162052856041983639777565, −19.57032837217219960177055084683, −19.24460634467503549568582535304, −18.03000054136306653879733777854, −17.49928556316922461322921634809, −16.25668706508120396367864421850, −15.56788133683405051941392217942, −14.99778366400730916027682596080, −13.75476187517244305520855470552, −12.36647881748777513031535367047, −11.74300788552470032498762298533, −10.832669105557022464444502300286, −10.166959686888111180714544657638, −9.19794034970815201171203798312, −8.67697519252932475797221246048, −7.286137556009011941431561176481, −6.51957288210706008173616184463, −4.85432779089962417829437978193, −4.003591853508876395873966495604, −2.778752278288893375446394616290, −2.319275972264867501018668700573, 0.25907661926225198786515278986, 1.04690637355333555985882247765, 2.478324916315255721193749992178, 4.05230055618144322372789962469, 5.4072058854741042346021299081, 6.208064272497266147022699324869, 7.232469673333220905060007568733, 8.04875656055926256131249020147, 8.500133160797582321306545663123, 9.716626233888044009239972399155, 10.7606694442871549723828234847, 11.66302863175852713668350314310, 12.87488922653280797429849265332, 13.454089475603263732050988393520, 14.40872380737227663100496511234, 15.58361287756280329887226919114, 16.36821491149316815692507211973, 17.14586924214234128730654712673, 17.68782583677344776245291686081, 18.86936544231933821216215202224, 19.48853036181213074600597677557, 20.034156293892482698513476645358, 20.906371931162351282114687136627, 22.73829137325842276517302203243, 23.16522667198678169613705698625

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