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Properties

Label 1-23e2-529.64-r0-0-0
Degree $1$
Conductor $529$
Sign $0.951 - 0.307i$
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  + (−0.358 + 0.933i)2-s + (0.940 − 0.340i)3-s + (−0.743 − 0.668i)4-s + (−0.691 + 0.722i)5-s + (−0.0186 + 0.999i)6-s + (0.901 − 0.432i)7-s + (0.890 − 0.454i)8-s + (0.767 − 0.640i)9-s + (−0.426 − 0.904i)10-s + (0.437 − 0.899i)11-s + (−0.926 − 0.375i)12-s + (−0.834 − 0.551i)13-s + (0.0806 + 0.996i)14-s + (−0.404 + 0.914i)15-s + (0.105 + 0.994i)16-s + (0.154 − 0.987i)17-s + ⋯
L(s)  = 1  + (−0.358 + 0.933i)2-s + (0.940 − 0.340i)3-s + (−0.743 − 0.668i)4-s + (−0.691 + 0.722i)5-s + (−0.0186 + 0.999i)6-s + (0.901 − 0.432i)7-s + (0.890 − 0.454i)8-s + (0.767 − 0.640i)9-s + (−0.426 − 0.904i)10-s + (0.437 − 0.899i)11-s + (−0.926 − 0.375i)12-s + (−0.834 − 0.551i)13-s + (0.0806 + 0.996i)14-s + (−0.404 + 0.914i)15-s + (0.105 + 0.994i)16-s + (0.154 − 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(529\)    =    \(23^{2}\)
Sign: $0.951 - 0.307i$
Analytic conductor: \(2.45666\)
Root analytic conductor: \(2.45666\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{529} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 529,\ (0:\ ),\ 0.951 - 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.291858481 - 0.2032163589i\)
\(L(\frac12)\) \(\approx\) \(1.291858481 - 0.2032163589i\)
\(L(1)\) \(\approx\) \(1.071109062 + 0.1294197464i\)
\(L(1)\) \(\approx\) \(1.071109062 + 0.1294197464i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
good2 \( 1 + (-0.358 + 0.933i)T \)
3 \( 1 + (0.940 - 0.340i)T \)
5 \( 1 + (-0.691 + 0.722i)T \)
7 \( 1 + (0.901 - 0.432i)T \)
11 \( 1 + (0.437 - 0.899i)T \)
13 \( 1 + (-0.834 - 0.551i)T \)
17 \( 1 + (0.154 - 0.987i)T \)
19 \( 1 + (-0.806 + 0.591i)T \)
29 \( 1 + (-0.117 - 0.993i)T \)
31 \( 1 + (-0.977 - 0.209i)T \)
37 \( 1 + (-0.860 - 0.508i)T \)
41 \( 1 + (0.988 + 0.148i)T \)
43 \( 1 + (0.980 + 0.197i)T \)
47 \( 1 + (-0.0682 - 0.997i)T \)
53 \( 1 + (-0.426 + 0.904i)T \)
59 \( 1 + (-0.873 + 0.487i)T \)
61 \( 1 + (0.813 + 0.581i)T \)
67 \( 1 + (0.999 + 0.0248i)T \)
71 \( 1 + (-0.759 - 0.650i)T \)
73 \( 1 + (-0.117 + 0.993i)T \)
79 \( 1 + (0.931 - 0.363i)T \)
83 \( 1 + (-0.191 + 0.981i)T \)
89 \( 1 + (0.735 + 0.678i)T \)
97 \( 1 + (0.912 + 0.409i)T \)
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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

−23.6611579054080721519272481923, −22.27661344660944594080822744300, −21.56748033525812675321057849195, −20.84518634861021888367759069763, −20.16019037465131543664664724682, −19.46128842092114740772330279232, −18.911519758633123453582290409803, −17.62661589890798021870252295827, −16.97831692274715065277968780017, −15.83042202681707886260688896225, −14.770975261582142929159232828931, −14.29417907449129629461899531033, −12.78389308750247874876720400976, −12.48230074025026538608033202282, −11.38281798471015981861207681234, −10.50165883351212188461716488569, −9.301416798108477413483766780186, −8.877626543320473733600211661847, −7.99747909051262361096527784592, −7.2413804481334758825382656325, −4.960173186276369714068359815231, −4.45901208789774809189504256487, −3.57196824197905622491170927220, −2.181000847318282187702700312737, −1.56042286614970837912894532761, 0.738878170379410736414667417, 2.256838225199698965268736358048, 3.61735393408253919118490947632, 4.44745424755569040878202610620, 5.81999038462633149392421978906, 7.01496205647390269298351849440, 7.6124529535822665325652274038, 8.21260793959710346535826813916, 9.159808403406301044658929436060, 10.24113740152657311394810403874, 11.15210378161649762900822620565, 12.366954379565053882453266877448, 13.643151775916184615480356915938, 14.32214672365622277129311943442, 14.78327243331490494042184259972, 15.605036425101414424060671736940, 16.62071392247402752033053521047, 17.62770336832894867605192402516, 18.42086435515041606453961523011, 19.15529906426172536800753975341, 19.745686438108006796522287209855, 20.76460877409026087059352970923, 21.90086391948008085461104948769, 22.916735101430767376276316218279, 23.64287789267977567505322085220

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