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Properties

Label 1-380-380.179-r0-0-0
Degree $1$
Conductor $380$
Sign $-0.0977 + 0.995i$
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  + (0.5 + 0.866i)3-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + 37-s − 39-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + 7-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)21-s + (−0.5 + 0.866i)23-s − 27-s + (0.5 − 0.866i)29-s + 31-s + (−0.5 − 0.866i)33-s + 37-s − 39-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(1.76471\)
Root analytic conductor: \(1.76471\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 380,\ (0:\ ),\ -0.0977 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9985469332 + 1.101414114i\)
\(L(\frac12)\) \(\approx\) \(0.9985469332 + 1.101414114i\)
\(L(1)\) \(\approx\) \(1.125424055 + 0.5265880996i\)
\(L(1)\) \(\approx\) \(1.125424055 + 0.5265880996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−24.47904951306422923948741384781, −23.54964341868518981145026229159, −22.84850780565141507478073555040, −21.52605114069605438961514164981, −20.61283042174087912558107340980, −20.099104059639672304605329126638, −18.933501452787400008929657153751, −18.087442513206531666088104353, −17.691534357733135823925585101530, −16.37393161773105993670900874526, −15.19381721214025402705603350548, −14.4459686011351487255263998410, −13.633459809395593327188218650552, −12.653423220642320408547227798968, −11.90189438190817660799333621532, −10.78389043446992618784070216473, −9.72310581217960274452875503355, −8.33493397681262955244567257665, −7.921617424108161460069457605, −6.955254924428459101122129900209, −5.63364363908334163256084597015, −4.69959527317927804477268840175, −3.05209381770639268853609787402, −2.25782515115234275064246433108, −0.857228889370875493380370613496, 1.79211575579976979282161887625, 2.86029339391760760580257990818, 4.20543738250054927714746077271, 4.89172638318723720511328821446, 5.99772390050135918828938292554, 7.73003521051976165109377864375, 8.15694923246287664479006867731, 9.38580776252938669769730084973, 10.22140476738393842540828517514, 11.11174084544591012334365932237, 12.03345787067305951451202556780, 13.4278335925368883101579035671, 14.21408119586869239700130556525, 15.02785894793563088987922451720, 15.77112672267565449063332471772, 16.80839220976855970601895182029, 17.6041306970742250585292136172, 18.77014530508709803154424609772, 19.625945591673370713289635416413, 20.60790262092550677643183308607, 21.41009400890981906631389304030, 21.698402825709960390533982099090, 23.13001366817435063867493269931, 23.890060747280243463042697279005, 24.82643430193547832733326311512

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