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Properties

Label 2-380-380.207-c1-0-29
Degree $2$
Conductor $380$
Sign $-0.431 - 0.902i$
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  + (−0.601 + 1.27i)2-s + (0.216 + 2.47i)3-s + (−1.27 − 1.53i)4-s + (1.88 − 1.20i)5-s + (−3.29 − 1.20i)6-s + (4.44 + 1.19i)7-s + (2.73 − 0.707i)8-s + (−3.10 + 0.546i)9-s + (0.402 + 3.13i)10-s + (1.86 + 1.07i)11-s + (3.52 − 3.48i)12-s + (0.0817 − 0.934i)13-s + (−4.19 + 4.96i)14-s + (3.37 + 4.40i)15-s + (−0.741 + 3.93i)16-s + (0.155 − 0.108i)17-s + ⋯
L(s)  = 1  + (−0.425 + 0.905i)2-s + (0.124 + 1.42i)3-s + (−0.638 − 0.769i)4-s + (0.843 − 0.537i)5-s + (−1.34 − 0.493i)6-s + (1.67 + 0.449i)7-s + (0.968 − 0.250i)8-s + (−1.03 + 0.182i)9-s + (0.127 + 0.991i)10-s + (0.562 + 0.324i)11-s + (1.01 − 1.00i)12-s + (0.0226 − 0.259i)13-s + (−1.12 + 1.32i)14-s + (0.871 + 1.13i)15-s + (−0.185 + 0.982i)16-s + (0.0376 − 0.0263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.431 - 0.902i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.431 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780345 + 1.23800i\)
\(L(\frac12)\) \(\approx\) \(0.780345 + 1.23800i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.601 - 1.27i)T \)
5 \( 1 + (-1.88 + 1.20i)T \)
19 \( 1 + (-0.561 + 4.32i)T \)
good3 \( 1 + (-0.216 - 2.47i)T + (-2.95 + 0.520i)T^{2} \)
7 \( 1 + (-4.44 - 1.19i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.86 - 1.07i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.0817 + 0.934i)T + (-12.8 - 2.25i)T^{2} \)
17 \( 1 + (-0.155 + 0.108i)T + (5.81 - 15.9i)T^{2} \)
23 \( 1 + (1.50 + 3.22i)T + (-14.7 + 17.6i)T^{2} \)
29 \( 1 + (4.99 - 0.880i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (7.09 - 4.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.09 + 3.09i)T + 37iT^{2} \)
41 \( 1 + (8.95 + 7.51i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.96 + 1.38i)T + (27.6 + 32.9i)T^{2} \)
47 \( 1 + (1.49 + 1.04i)T + (16.0 + 44.1i)T^{2} \)
53 \( 1 + (9.89 - 4.61i)T + (34.0 - 40.6i)T^{2} \)
59 \( 1 + (0.0187 - 0.106i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.61 - 1.31i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-4.93 - 3.45i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (-3.03 - 8.33i)T + (-54.3 + 45.6i)T^{2} \)
73 \( 1 + (-10.0 + 0.877i)T + (71.8 - 12.6i)T^{2} \)
79 \( 1 + (-1.87 - 1.57i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.13 - 15.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.23 + 1.46i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-13.6 + 9.58i)T + (33.1 - 91.1i)T^{2} \)
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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

−11.19069434527910820253886673205, −10.52353648648665321274891829514, −9.547098381795909753117075367355, −8.916987385971296694770736317084, −8.328847463392506655428541743481, −6.95898911798768020928086520934, −5.36933210929720709704868523083, −5.13977286821973529697728702719, −4.12455930036387250468775568168, −1.78647687837391969961682052007, 1.54488102130559737724601448115, 1.87569966319169511388203635931, 3.59339283818775784454055292980, 5.17649742124179719716648691446, 6.51541810644674662245314936594, 7.64224129103913151740193946195, 8.076812625486288215312374699640, 9.245974132948280861990376818466, 10.29535819200336161418268455360, 11.36820770634207611512078767673

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