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Properties

Label 1-148-148.59-r0-0-0
Degree $1$
Conductor $148$
Sign $0.587 - 0.809i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.587 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(148\)    =    \(2^{2} \cdot 37\)
Sign: $0.587 - 0.809i$
Analytic conductor: \(0.687309\)
Root analytic conductor: \(0.687309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{148} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 148,\ (0:\ ),\ 0.587 - 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337338843 - 0.6821293593i\)
\(L(\frac12)\) \(\approx\) \(1.337338843 - 0.6821293593i\)
\(L(1)\) \(\approx\) \(1.322943151 - 0.4157518168i\)
\(L(1)\) \(\approx\) \(1.322943151 - 0.4157518168i\)

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

−28.02094227782048017985639186110, −26.919860825399791994847106362296, −26.6405820720343622280899098181, −25.46318587520297447498199786001, −24.57854042731878681539757188125, −23.35106431094244895296750577985, −22.09243937058234645012333388517, −21.35660464450508097059959745986, −20.57617167525102263356205170064, −19.29723615683470601487874055424, −18.49057312746253934314797875723, −17.23199021977309386060862792979, −16.14111950960814453755923244856, −14.71561316174031278000837827486, −14.45151947932005143344227510442, −13.373510358930373369390013691265, −11.64195434780264733801714435660, −10.460519386920071838800195505207, −9.91251631137921385778099656324, −8.2584993368357699494465141053, −7.59766911244892098240543092983, −5.88336327526985602801573462873, −4.551919033932538683373849515861, −3.21761307000961741002896557242, −2.124545131194434211605862981407, 1.50218192358748140014316913481, 2.48984345212292648094744057252, 4.41128002661309037859312632130, 5.47660570632765846490834323961, 7.17081026001450128547661827233, 8.1333097244689606040195750266, 9.048634012669462907172186640159, 10.17089401158225473641997570470, 12.08473076926800661985655345885, 12.519215378882165475087433479622, 13.81496906984154414984320100866, 14.662887085388805261996417724010, 15.72976063321440736266515916206, 17.33478418164140594147830313376, 17.84286568346725795120101555191, 19.15726228481158259099003021390, 20.12565402166468820677635426277, 20.93021381262198769638467419641, 21.75113101570139118586470443077, 23.672140997954245410226961025866, 23.99648872397924807948751925977, 25.184288123823299012515526384433, 25.657425952635136216388545214641, 27.065721667665420460052548255, 28.04328637911616859987678779346

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