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Properties

Label 2-380-95.87-c0-0-0
Degree $2$
Conductor $380$
Sign $0.720 - 0.693i$
Analytic cond. $0.189644$
Root an. cond. $0.435482$
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  + (0.366 + 1.36i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)9-s + 11-s + (−1.36 − 0.366i)13-s + (1.36 + 0.366i)15-s + (−1.36 + 0.366i)17-s + (−0.866 + 0.5i)19-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s − 31-s + (0.366 + 1.36i)33-s − 2i·39-s + 0.999i·45-s + (0.366 − 1.36i)47-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)9-s + 11-s + (−1.36 − 0.366i)13-s + (1.36 + 0.366i)15-s + (−1.36 + 0.366i)17-s + (−0.866 + 0.5i)19-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s − 31-s + (0.366 + 1.36i)33-s − 2i·39-s + 0.999i·45-s + (0.366 − 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.720 - 0.693i$
Analytic conductor: \(0.189644\)
Root analytic conductor: \(0.435482\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :0),\ 0.720 - 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9487688219\)
\(L(\frac12)\) \(\approx\) \(0.9487688219\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - T + T^{2} \)
13 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.64528581294969500628567482985, −10.39623782197591880128470206888, −9.896923226601249558597266371690, −8.958989176245279711773132789448, −8.515009411850474889635453423196, −6.91422105091519552427498478885, −5.62582439184361151769534193784, −4.59441233697527611659881268374, −3.95301329207217017704523699620, −2.25265492427407249224825261697, 1.90934274220087616238635268671, 2.75415745285913198130326056060, 4.49820571930749815801506150033, 6.15120767209249803948321833368, 6.91734838173258595631344649933, 7.32910943188682190145470721728, 8.707044893252348324896354461413, 9.468723461770048260052929318076, 10.67890830444181430261363525751, 11.59203570801230045110270138800

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