Properties

Label 1-145-145.124-r1-0-0
Degree $1$
Conductor $145$
Sign $-0.191 - 0.981i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (−0.781 − 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.433 − 0.900i)11-s + i·12-s + (−0.900 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s i·17-s + (−0.433 − 0.900i)18-s + (0.974 − 0.222i)19-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.974 + 0.222i)3-s + (0.222 + 0.974i)4-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (−0.433 − 0.900i)11-s + i·12-s + (−0.900 + 0.433i)13-s + (−0.781 + 0.623i)14-s + (−0.900 + 0.433i)16-s i·17-s + (−0.433 − 0.900i)18-s + (0.974 − 0.222i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.191 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $-0.191 - 0.981i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ -0.191 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9733753531 - 1.181861528i\)
\(L(\frac12)\) \(\approx\) \(0.9733753531 - 1.181861528i\)
\(L(1)\) \(\approx\) \(0.9419793395 - 0.4234566163i\)
\(L(1)\) \(\approx\) \(0.9419793395 - 0.4234566163i\)

Euler product

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

−28.05765924886597185799482144382, −27.144518295324040175925579518348, −26.10952712751499934286285344896, −25.43218341263275209350944155381, −24.58581862642541538846370543093, −23.87115416663729503409652861499, −22.39534074429788432794067526446, −21.04579153559025264409668644481, −20.00651174563216068538001538863, −19.21851057026101700922404504804, −18.20114174220995839637345711569, −17.50154719028609537580095293676, −15.88759449424032463785545434895, −15.11619181509646246997037719177, −14.47263808688728068216584854782, −13.03786932315514061110213468448, −11.823133381578750057822924445152, −10.04963827405288152621736755993, −9.44313836714599393162865032314, −8.12529723039215114226349627911, −7.59320934409067098340882254652, −6.12741500164932983783297295457, −4.77793888128436664131269699040, −2.714017102351454114363269200112, −1.597145667690600265987488002443, 0.68781964116709886656048842945, 2.32770011549537753983492356489, 3.41962775878110070751691357773, 4.654821752795807759514201204532, 7.09381838467155207017553381410, 7.83848647448667886659551497813, 8.98210747396332124253036381898, 9.95660670502178820846515139525, 10.851815966409663665135373015683, 12.145625263269061195480704681944, 13.546397601795025668356195649341, 14.16884490848948851989255850004, 15.86780718694363874544590978157, 16.5854854858245001415407706372, 17.89686323087754242298040684781, 18.917380817418428766381292993368, 19.81301760365933205883643130184, 20.541037672699418159814320077240, 21.37615513426824035079598816142, 22.41978187573138826425259968637, 24.14171095934178193235847087001, 24.920097319370998551336309529529, 26.30174799882315224909263614210, 26.664563386727765360318037522795, 27.36375889519271227073520470014

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