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Properties

Label 1-380-380.83-r0-0-0
Degree $1$
Conductor $380$
Sign $0.240 + 0.970i$
Analytic cond. $1.76471$
Root an. cond. $1.76471$
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  + (0.866 + 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s − 11-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − 31-s + (−0.866 − 0.5i)33-s − i·37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s − i·7-s + (0.5 + 0.866i)9-s − 11-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + (−0.5 + 0.866i)21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − 31-s + (−0.866 − 0.5i)33-s − i·37-s + 39-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.240 + 0.970i$
Analytic conductor: \(1.76471\)
Root analytic conductor: \(1.76471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 380,\ (0:\ ),\ 0.240 + 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.325571489 + 1.037270388i\)
\(L(\frac12)\) \(\approx\) \(1.325571489 + 1.037270388i\)
\(L(1)\) \(\approx\) \(1.284237479 + 0.4710426233i\)
\(L(1)\) \(\approx\) \(1.284237479 + 0.4710426233i\)

Euler product

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

−24.20441693877735531039123614759, −23.6534886271499496205379284264, −22.9190483180481388958605908015, −21.48178727272734993795289745553, −20.58221250843384485835544633495, −20.25496810798869507678194544441, −18.94695517783804484420229053704, −18.49753919343395931380570894105, −17.42678977970448649521081508757, −16.28728619826077885411835936689, −15.5063096586582174268182823943, −14.229151257124927452354962315921, −13.76870358366127141275152179730, −12.9290291813745014237793558543, −11.89032550248701773259842698001, −10.61665347832290341878687308682, −9.821847842438648324616285955012, −8.63733891172951247826191275158, −7.77545677039884209341993355776, −7.04809487293726703385233431419, −5.86686548343659045934225275159, −4.34761401597048344564924098284, −3.43859983403546943082463230604, −2.269479182870793487503597376394, −0.96482342486647219016967192351, 1.77682099089218602735433346820, 2.87924630253780592297174855926, 3.74145494909630032665624152870, 5.14836444485685806123195948439, 5.91011362597027199368031535273, 7.58314801876064812563365943390, 8.305942957805968313589461917836, 9.137514236963547259252645528149, 10.16983633235877352833500130782, 10.96327096168958001049620325692, 12.349506791711648850998204607316, 13.1282841928765916775371809632, 14.15457747856712085975098373958, 15.0548530833781524498757813158, 15.764301686439229554708095585636, 16.428245543803706936399034265708, 18.03616148516466979762825422402, 18.545994301036960763179097528195, 19.57345130182305305911767859205, 20.42901752363812642205458307582, 21.35727356689438194620761674648, 21.743344600464394977012030612302, 22.98958995805454745880914192381, 23.93174296543481876892421278930, 24.97878010493135399001272542450

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