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Properties

Label 2-380-20.7-c1-0-29
Degree $2$
Conductor $380$
Sign $-0.449 + 0.893i$
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  + (−1.38 − 0.291i)2-s + (−1.18 − 1.18i)3-s + (1.83 + 0.805i)4-s + (1.27 − 1.83i)5-s + (1.29 + 1.98i)6-s + (1.53 − 1.53i)7-s + (−2.29 − 1.64i)8-s − 0.182i·9-s + (−2.29 + 2.17i)10-s + 3.80i·11-s + (−1.21 − 3.12i)12-s + (4.33 − 4.33i)13-s + (−2.57 + 1.67i)14-s + (−3.69 + 0.672i)15-s + (2.70 + 2.94i)16-s + (3.54 + 3.54i)17-s + ⋯
L(s)  = 1  + (−0.978 − 0.205i)2-s + (−0.685 − 0.685i)3-s + (0.915 + 0.402i)4-s + (0.568 − 0.822i)5-s + (0.529 + 0.811i)6-s + (0.580 − 0.580i)7-s + (−0.812 − 0.582i)8-s − 0.0607i·9-s + (−0.726 + 0.687i)10-s + 1.14i·11-s + (−0.351 − 0.903i)12-s + (1.20 − 1.20i)13-s + (−0.688 + 0.448i)14-s + (−0.953 + 0.173i)15-s + (0.675 + 0.737i)16-s + (0.858 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.427736 - 0.694161i\)
\(L(\frac12)\) \(\approx\) \(0.427736 - 0.694161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.291i)T \)
5 \( 1 + (-1.27 + 1.83i)T \)
19 \( 1 + T \)
good3 \( 1 + (1.18 + 1.18i)T + 3iT^{2} \)
7 \( 1 + (-1.53 + 1.53i)T - 7iT^{2} \)
11 \( 1 - 3.80iT - 11T^{2} \)
13 \( 1 + (-4.33 + 4.33i)T - 13iT^{2} \)
17 \( 1 + (-3.54 - 3.54i)T + 17iT^{2} \)
23 \( 1 + (3.11 + 3.11i)T + 23iT^{2} \)
29 \( 1 - 3.00iT - 29T^{2} \)
31 \( 1 + 8.41iT - 31T^{2} \)
37 \( 1 + (8.05 + 8.05i)T + 37iT^{2} \)
41 \( 1 + 2.12T + 41T^{2} \)
43 \( 1 + (-2.98 - 2.98i)T + 43iT^{2} \)
47 \( 1 + (-2.50 + 2.50i)T - 47iT^{2} \)
53 \( 1 + (6.45 - 6.45i)T - 53iT^{2} \)
59 \( 1 - 9.56T + 59T^{2} \)
61 \( 1 + 0.367T + 61T^{2} \)
67 \( 1 + (6.11 - 6.11i)T - 67iT^{2} \)
71 \( 1 + 4.48iT - 71T^{2} \)
73 \( 1 + (-0.985 + 0.985i)T - 73iT^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + (-11.9 - 11.9i)T + 97iT^{2} \)
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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

−10.83146444701022092932351861199, −10.29279589975442644005199479209, −9.234190331727208064136036251072, −8.193917879396597207556841177182, −7.52071189965067189920133231470, −6.31967809310606282719771942157, −5.57284712137603157501709705689, −3.91591575220882888269446094553, −1.85314887428797727742322605216, −0.874676112426092752733333102816, 1.77190622067395851058144818240, 3.34123185055669853812488803983, 5.23298322757961545347288972103, 5.95101563424507369127860713443, 6.82309694959213856496574075161, 8.144455969413170954738179977984, 8.961738211854283643020403800362, 9.948835979196728424954993191782, 10.67031432655754961149166838271, 11.44603733261065263554241881157

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