Properties

Label 1-293-293.57-r0-0-0
Degree $1$
Conductor $293$
Sign $0.474 - 0.880i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (0.869 − 0.493i)2-s + (0.823 + 0.566i)3-s + (0.512 − 0.858i)4-s + (0.192 − 0.981i)5-s + (0.996 + 0.0859i)6-s + (−0.999 + 0.0430i)7-s + (0.0215 − 0.999i)8-s + (0.357 + 0.933i)9-s + (−0.317 − 0.948i)10-s + (−0.954 − 0.296i)11-s + (0.908 − 0.417i)12-s + (0.985 + 0.171i)13-s + (−0.847 + 0.530i)14-s + (0.714 − 0.699i)15-s + (−0.474 − 0.880i)16-s + (0.869 − 0.493i)17-s + ⋯
L(s)  = 1  + (0.869 − 0.493i)2-s + (0.823 + 0.566i)3-s + (0.512 − 0.858i)4-s + (0.192 − 0.981i)5-s + (0.996 + 0.0859i)6-s + (−0.999 + 0.0430i)7-s + (0.0215 − 0.999i)8-s + (0.357 + 0.933i)9-s + (−0.317 − 0.948i)10-s + (−0.954 − 0.296i)11-s + (0.908 − 0.417i)12-s + (0.985 + 0.171i)13-s + (−0.847 + 0.530i)14-s + (0.714 − 0.699i)15-s + (−0.474 − 0.880i)16-s + (0.869 − 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ 0.474 - 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.156356994 - 1.287762698i\)
\(L(\frac12)\) \(\approx\) \(2.156356994 - 1.287762698i\)
\(L(1)\) \(\approx\) \(1.896955389 - 0.6826513252i\)
\(L(1)\) \(\approx\) \(1.896955389 - 0.6826513252i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.869 - 0.493i)T \)
3 \( 1 + (0.823 + 0.566i)T \)
5 \( 1 + (0.192 - 0.981i)T \)
7 \( 1 + (-0.999 + 0.0430i)T \)
11 \( 1 + (-0.954 - 0.296i)T \)
13 \( 1 + (0.985 + 0.171i)T \)
17 \( 1 + (0.869 - 0.493i)T \)
19 \( 1 + (0.714 - 0.699i)T \)
23 \( 1 + (0.996 - 0.0859i)T \)
29 \( 1 + (-0.798 + 0.601i)T \)
31 \( 1 + (0.192 + 0.981i)T \)
37 \( 1 + (-0.548 + 0.835i)T \)
41 \( 1 + (-0.991 - 0.128i)T \)
43 \( 1 + (-0.0645 + 0.997i)T \)
47 \( 1 + (-0.976 - 0.213i)T \)
53 \( 1 + (-0.954 - 0.296i)T \)
59 \( 1 + (0.651 + 0.758i)T \)
61 \( 1 + (0.584 + 0.811i)T \)
67 \( 1 + (-0.150 - 0.988i)T \)
71 \( 1 + (-0.618 + 0.785i)T \)
73 \( 1 + (0.512 + 0.858i)T \)
79 \( 1 + (0.584 + 0.811i)T \)
83 \( 1 + (-0.234 + 0.972i)T \)
89 \( 1 + (0.584 - 0.811i)T \)
97 \( 1 + (0.436 - 0.899i)T \)
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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

−25.573673532666844508528851069520, −24.92319499197549201160677805440, −23.57456674871103127103581712429, −23.103173586206168170616830487750, −22.24294368355661260256185708688, −21.0077954771365757832056170823, −20.51561437363729178401320601432, −19.07883654559078301392281133363, −18.56508207480653469335930843096, −17.40625899985605892305097292309, −16.07749008502979194854615612843, −15.287672084612656191601765809282, −14.538164571353492511821786913004, −13.47114660946610026419046785293, −13.09172034434514159942397945112, −11.991304671581668056279818765687, −10.64923075174514278447573535258, −9.54276709804249525963600311636, −8.08705612681154772119732173256, −7.36895919785549753530409342606, −6.43051656062811932963676477680, −5.60719471104175878257047499430, −3.611915503046440578408370287855, −3.234741463058664839106172306298, −2.049936680130581279151151924151, 1.31784106411629252451381838520, 2.8934960368183088820094640226, 3.48692187131337010818893926851, 4.86058030234919709465695511038, 5.5155184156866021509529753771, 6.992683028163176921101414466235, 8.4633529884706706480990656145, 9.433956400157975860571982186273, 10.17198409642124371107156774432, 11.30623686200044784615132576001, 12.653797731749479817001878222146, 13.31657949760443784048081530585, 13.87546085769251691659497325679, 15.21690730505977977520614600650, 16.05187252620963074127287529420, 16.440899245425375645149061012512, 18.44543987336153294806340119799, 19.29260637376416654713371604120, 20.186299344728031541224730415867, 20.87058799960808550395059054285, 21.41539569744143906593307915224, 22.48558444110104894963318661021, 23.42991170992834360316190075836, 24.34587135785502712060782953480, 25.33609012812549907068976008071

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