Properties

Label 2-380-380.379-c1-0-50
Degree $2$
Conductor $380$
Sign $-0.882 + 0.469i$
Analytic cond. $3.03431$
Root an. cond. $1.74192$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.948 − 1.04i)2-s − 1.76i·3-s + (−0.201 − 1.98i)4-s + (−1.85 − 1.24i)5-s + (−1.84 − 1.67i)6-s + 4.29·7-s + (−2.27 − 1.67i)8-s − 0.102·9-s + (−3.06 + 0.770i)10-s + 0.376i·11-s + (−3.50 + 0.355i)12-s − 0.928·13-s + (4.07 − 4.50i)14-s + (−2.19 + 3.27i)15-s + (−3.91 + 0.802i)16-s + 5.96i·17-s + ⋯
L(s)  = 1  + (0.670 − 0.741i)2-s − 1.01i·3-s + (−0.100 − 0.994i)4-s + (−0.831 − 0.556i)5-s + (−0.754 − 0.681i)6-s + 1.62·7-s + (−0.805 − 0.592i)8-s − 0.0343·9-s + (−0.969 + 0.243i)10-s + 0.113i·11-s + (−1.01 + 0.102i)12-s − 0.257·13-s + (1.08 − 1.20i)14-s + (−0.565 + 0.845i)15-s + (−0.979 + 0.200i)16-s + 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447292 - 1.79313i\)
\(L(\frac12)\) \(\approx\) \(0.447292 - 1.79313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.948 + 1.04i)T \)
5 \( 1 + (1.85 + 1.24i)T \)
19 \( 1 + (3.61 + 2.43i)T \)
good3 \( 1 + 1.76iT - 3T^{2} \)
7 \( 1 - 4.29T + 7T^{2} \)
11 \( 1 - 0.376iT - 11T^{2} \)
13 \( 1 + 0.928T + 13T^{2} \)
17 \( 1 - 5.96iT - 17T^{2} \)
23 \( 1 + 0.352T + 23T^{2} \)
29 \( 1 - 3.89iT - 29T^{2} \)
31 \( 1 - 1.06T + 31T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 + 6.82iT - 41T^{2} \)
43 \( 1 - 9.45T + 43T^{2} \)
47 \( 1 - 2.56T + 47T^{2} \)
53 \( 1 + 3.18T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 + 3.22T + 61T^{2} \)
67 \( 1 + 8.75iT - 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 5.09iT - 73T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + 3.49T + 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 - 9.74T + 97T^{2} \)
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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

−11.19245089111107027929742476601, −10.55116617421182597375051151605, −8.968450625483151968548392146402, −8.112851195648454550429736833625, −7.28553219898743533568985199332, −5.99462093233216354153346762856, −4.74716442585100593951853929198, −4.08586908589233607928646067736, −2.17510838650890760369633280238, −1.15389890978515602184687857747, 2.78379297491701036088321362502, 4.35931643000455790342322213662, 4.45475093945461581321461198259, 5.75166703752346363967328769764, 7.19702898043279895350040979453, 7.84733398005350289633280478758, 8.722116034811389573849148339866, 9.952358294082762824995133791629, 11.21671533139360076524869090382, 11.46157149307812674835216396509

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