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Properties

Label 1-259-259.55-r0-0-0
Degree $1$
Conductor $259$
Sign $0.580 - 0.814i$
Analytic cond. $1.20279$
Root an. cond. $1.20279$
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  + (−0.984 + 0.173i)2-s + (0.173 − 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (−0.866 + 0.5i)8-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)18-s + (0.984 + 0.173i)19-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.173 − 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (−0.866 + 0.5i)8-s + (−0.939 − 0.342i)9-s + (0.5 − 0.866i)10-s + (0.5 + 0.866i)11-s + (−0.173 − 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (0.984 + 0.173i)18-s + (0.984 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.580 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(259\)    =    \(7 \cdot 37\)
Sign: $0.580 - 0.814i$
Analytic conductor: \(1.20279\)
Root analytic conductor: \(1.20279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{259} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 259,\ (0:\ ),\ 0.580 - 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6349185261 - 0.3272149525i\)
\(L(\frac12)\) \(\approx\) \(0.6349185261 - 0.3272149525i\)
\(L(1)\) \(\approx\) \(0.6725396161 - 0.1506540698i\)
\(L(1)\) \(\approx\) \(0.6725396161 - 0.1506540698i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.642 + 0.766i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.342 - 0.939i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.642 + 0.766i)T \)
61 \( 1 + (0.342 + 0.939i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.642 - 0.766i)T \)
83 \( 1 + (0.939 + 0.342i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−26.49209956337465727023094981382, −25.28174209616514826971284064778, −24.44173176080987803778425447823, −23.476844022010283354657068128698, −21.96636145669754095663557644354, −21.30722564382436929426509724328, −20.41501566264838594592104830765, −19.56714456435378475330812103177, −19.0097200713609725067119110239, −17.44744121494000067242781922201, −16.57628649161882636299042830500, −16.18033717894351193735217574830, −15.15713602063632843068392656094, −14.06761228439150908994654419881, −12.42064692390220601091488086653, −11.548475932063850930936490842876, −10.74461658360437556586610760357, −9.55416288971805424729821328218, −8.80772447185290442798952187137, −8.13878617718124604192574275462, −6.73143049855647794324895728388, −5.27468629027919811177929889125, −3.98702828320075081979358892933, −3.00371780180876534613823607974, −1.194223092650838709138727174495, 0.80502696929834303394558027847, 2.330283536112933840958349473869, 3.28331784965412953219141366647, 5.43614265932334793836267575175, 6.783072888496427852007629071583, 7.35567037922690904669525745855, 8.08395846486014850818335805451, 9.37086306867638548198923801611, 10.38498872564122891362799384482, 11.71341559874288766780636246884, 12.0469287637724291139391175995, 13.625686519082864897616600028804, 14.81476718274137931815597691719, 15.36601805820892029209308886996, 16.72922944964456904979457226536, 17.76906965667570185163714654973, 18.296412518257024123054376368, 19.21965702934418344815311479603, 19.9277031410434585727288761960, 20.6478808534554257662753659447, 22.49209874862115383323640866467, 23.11885112806391086110806811842, 24.17149632800136168010156437255, 25.1636205042137210415793412145, 25.53769663411466833371436144500

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