Properties

Label 2-380-380.239-c2-0-53
Degree $2$
Conductor $380$
Sign $0.737 + 0.675i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.97 + 0.339i)2-s + (−1.97 + 3.42i)3-s + (3.76 − 1.33i)4-s + (−0.693 + 4.95i)5-s + (2.73 − 7.41i)6-s − 2.56·7-s + (−6.97 + 3.91i)8-s + (−3.30 − 5.73i)9-s + (−0.313 − 9.99i)10-s − 17.7i·11-s + (−2.87 + 15.5i)12-s + (−4.58 + 2.64i)13-s + (5.06 − 0.871i)14-s + (−15.5 − 12.1i)15-s + (12.4 − 10.0i)16-s + (1.90 + 1.09i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.169i)2-s + (−0.658 + 1.14i)3-s + (0.942 − 0.334i)4-s + (−0.138 + 0.990i)5-s + (0.455 − 1.23i)6-s − 0.367·7-s + (−0.872 + 0.489i)8-s + (−0.367 − 0.636i)9-s + (−0.0313 − 0.999i)10-s − 1.61i·11-s + (−0.239 + 1.29i)12-s + (−0.352 + 0.203i)13-s + (0.361 − 0.0622i)14-s + (−1.03 − 0.810i)15-s + (0.776 − 0.630i)16-s + (0.111 + 0.0645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.737 + 0.675i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.267891 - 0.104070i\)
\(L(\frac12)\) \(\approx\) \(0.267891 - 0.104070i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.97 - 0.339i)T \)
5 \( 1 + (0.693 - 4.95i)T \)
19 \( 1 + (15.3 + 11.2i)T \)
good3 \( 1 + (1.97 - 3.42i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + 2.56T + 49T^{2} \)
11 \( 1 + 17.7iT - 121T^{2} \)
13 \( 1 + (4.58 - 2.64i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-1.90 - 1.09i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (8.21 + 14.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.5 - 42.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 13.0iT - 961T^{2} \)
37 \( 1 + 53.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12.6 + 21.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-36.7 + 63.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-27.9 - 48.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-38.6 + 22.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-58.4 - 33.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.07 + 3.59i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (51.0 + 88.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-35.9 - 20.7i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-41.1 - 23.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (63.0 + 36.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 124.T + 6.88e3T^{2} \)
89 \( 1 + (-50.2 - 86.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (53.4 + 30.8i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.68975012825343764371163229410, −10.44447169550716467125836076423, −9.280927645216978989732949664351, −8.504982253882511561041900674834, −7.19277777047167522276348170413, −6.29133451102402134027433416396, −5.45960168666610932010282385287, −3.84068551798894972818445458236, −2.67141971356831567595385791158, −0.21154728777262209924801660770, 1.16248275264023286686096536675, 2.24954665382971656231802290847, 4.31157752560126819427671219040, 5.81261644311891404105308232837, 6.70839574549013765396686812699, 7.62088825172476162454771790641, 8.250817680039388460127124996408, 9.611979215646639614640927104987, 10.03613953134823654579345998875, 11.55163336332985712620018723056

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