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Properties

Label 1-148-148.15-r0-0-0
Degree $1$
Conductor $148$
Sign $0.999 + 0.0403i$
Analytic cond. $0.687309$
Root an. cond. $0.687309$
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  + (0.766 − 0.642i)3-s + (−0.342 + 0.939i)5-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (0.342 + 0.939i)15-s + (−0.984 − 0.173i)17-s + (0.642 + 0.766i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (−0.766 − 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (−0.342 + 0.939i)5-s + (0.939 + 0.342i)7-s + (0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (0.342 + 0.939i)15-s + (−0.984 − 0.173i)17-s + (0.642 + 0.766i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (−0.766 − 0.642i)25-s + (−0.5 − 0.866i)27-s + (−0.866 − 0.5i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.999 + 0.0403i$
Analytic conductor: \(0.687309\)
Root analytic conductor: \(0.687309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 148,\ (0:\ ),\ 0.999 + 0.0403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.424400670 + 0.02874372793i\)
\(L(\frac12)\) \(\approx\) \(1.424400670 + 0.02874372793i\)
\(L(1)\) \(\approx\) \(1.324590949 + 0.01708313827i\)
\(L(1)\) \(\approx\) \(1.324590949 + 0.01708313827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (-0.342 + 0.939i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (-0.984 - 0.173i)T \)
19 \( 1 + (0.642 + 0.766i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.939 + 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
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

−28.00069557667531108320460309155, −27.061692880316060598388419468870, −26.42923368613594547568910661760, −25.15919710104855874222498855077, −24.236243217627056009962207234710, −23.51548752422532931280807397647, −21.87732579966151935642434374221, −21.03985213697993164801958789130, −20.36311441858752369611371751752, −19.48856327400749520986212760574, −18.18376728103079732010353879509, −16.87005003616050524376704683723, −15.94569572224789003397753245579, −15.16226472364089991055384710418, −13.75983598677692020287805473258, −13.224054202398360621552344911445, −11.42896031954789537663259929672, −10.7141508515129669135618990995, −9.01696231160891838727317638241, −8.55561735773124631536858408774, −7.426319711531525790530404691252, −5.36789701141913030808723472423, −4.463313542279449411551218679836, −3.28572952393999318160117894158, −1.49337150528905872150180721877, 1.78285579154032777036219471758, 2.88948646607384137708471005510, 4.27950856880398208136372033243, 6.05729993866503297749199110370, 7.33700591876534553343575451796, 8.03014181563563600311448314335, 9.26224504746678183270271829368, 10.74229250090494082078303796125, 11.71303460496478228675965349240, 12.95766345326454089711012454936, 14.036851566575071571541858204220, 14.96367456970123321244331024993, 15.60081055566421778573040558278, 17.58022033750093027153564520440, 18.34543995633244444548024989441, 18.93390177245816475575197771564, 20.38145917274717317828767419686, 20.8681519952655274706416332739, 22.4088841897396640588312062831, 23.30828864110016830469365971701, 24.324452109827303161397048678881, 25.24566230594052201038471554558, 26.17940308905170648000326329573, 26.94811118333235677151564040058, 28.11479394793465706487051627532

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