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Properties

Label 1-145-145.34-r0-0-0
Degree $1$
Conductor $145$
Sign $0.831 + 0.556i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (−0.222 + 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 − 0.974i)6-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + 12-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + 17-s + (0.623 + 0.781i)18-s + (0.900 + 0.433i)19-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 + 0.433i)3-s + (−0.900 − 0.433i)4-s + (−0.222 − 0.974i)6-s + (0.900 − 0.433i)7-s + (0.623 − 0.781i)8-s + (0.623 − 0.781i)9-s + (−0.623 − 0.781i)11-s + 12-s + (−0.623 − 0.781i)13-s + (0.222 + 0.974i)14-s + (0.623 + 0.781i)16-s + 17-s + (0.623 + 0.781i)18-s + (0.900 + 0.433i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.831 + 0.556i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.831 + 0.556i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6732767000 + 0.2044851760i\)
\(L(\frac12)\) \(\approx\) \(0.6732767000 + 0.2044851760i\)
\(L(1)\) \(\approx\) \(0.6856720671 + 0.2539945700i\)
\(L(1)\) \(\approx\) \(0.6856720671 + 0.2539945700i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (-0.623 - 0.781i)T \)
13 \( 1 + (-0.623 - 0.781i)T \)
17 \( 1 + T \)
19 \( 1 + (0.900 + 0.433i)T \)
23 \( 1 + (0.222 + 0.974i)T \)
31 \( 1 + (0.222 - 0.974i)T \)
37 \( 1 + (0.623 - 0.781i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.900 - 0.433i)T \)
67 \( 1 + (-0.623 + 0.781i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.623 + 0.781i)T \)
83 \( 1 + (0.900 + 0.433i)T \)
89 \( 1 + (0.222 - 0.974i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−28.48085387975856517124292627285, −27.46081520298464016742454758251, −26.57546373107717419949525113104, −25.133897341921026109108943598, −23.94164257365627372781953419992, −23.12582227427405506593859737409, −22.0442761175390510330083965085, −21.28488663717917146548670313589, −20.24531142103140437805498663497, −18.89659793269281191833242438046, −18.24430081285213963657460189888, −17.429606852756857845723956291237, −16.41877307417731875843081487500, −14.74872666145142681749381883664, −13.53974101659140627505051440207, −12.26541710744593750192387737207, −11.838104034660492961772267879, −10.69288518667594008418816124351, −9.709134559688918607280601016228, −8.212794839336001131854331698622, −7.15098228193428270870191622998, −5.27285739928059949109876701013, −4.60254728348738675975143182299, −2.54741192188497095640903627874, −1.34814451645547212719105024127, 0.9055911157888081242952812359, 3.74651415918692782345728241795, 5.21339161124937516516081047118, 5.630668169781022636686622574131, 7.28029942895758757328809976948, 8.09288328017664226138246134337, 9.68815160873421003421561109606, 10.527346408651972967391765018638, 11.721179720267724037386452756228, 13.16487121700505107216456151787, 14.363293777301548191628998488008, 15.351159667746898252980579726324, 16.33341637814805650140956290162, 17.1804264232159656729175962319, 17.94807193021053944524389743262, 18.90864606602204887449010183638, 20.56459624020609219252326038769, 21.64209985731103384615208639009, 22.629199408148168465932088556014, 23.60284700203732014503501909956, 24.17935854276037916256241242583, 25.31617255696348628854492093381, 26.736534080527678275904441105720, 27.1295064706915210747131775850, 27.99330809577246167539800321818

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