Properties

Label 2-380-76.27-c1-0-15
Degree $2$
Conductor $380$
Sign $0.594 - 0.804i$
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  + (−0.785 + 1.17i)2-s + (1.08 − 1.87i)3-s + (−0.767 − 1.84i)4-s + (−0.5 + 0.866i)5-s + (1.35 + 2.74i)6-s + 3.94i·7-s + (2.77 + 0.548i)8-s + (−0.844 − 1.46i)9-s + (−0.626 − 1.26i)10-s + 3.27i·11-s + (−4.29 − 0.561i)12-s + (2.48 − 1.43i)13-s + (−4.64 − 3.09i)14-s + (1.08 + 1.87i)15-s + (−2.82 + 2.83i)16-s + (2.44 − 4.24i)17-s + ⋯
L(s)  = 1  + (−0.555 + 0.831i)2-s + (0.625 − 1.08i)3-s + (−0.383 − 0.923i)4-s + (−0.223 + 0.387i)5-s + (0.553 + 1.12i)6-s + 1.49i·7-s + (0.981 + 0.193i)8-s + (−0.281 − 0.487i)9-s + (−0.197 − 0.401i)10-s + 0.986i·11-s + (−1.23 − 0.162i)12-s + (0.689 − 0.398i)13-s + (−1.24 − 0.828i)14-s + (0.279 + 0.484i)15-s + (−0.705 + 0.708i)16-s + (0.593 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08061 + 0.544871i\)
\(L(\frac12)\) \(\approx\) \(1.08061 + 0.544871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.785 - 1.17i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-3.58 - 2.47i)T \)
good3 \( 1 + (-1.08 + 1.87i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.94iT - 7T^{2} \)
11 \( 1 - 3.27iT - 11T^{2} \)
13 \( 1 + (-2.48 + 1.43i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.44 + 4.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.53 - 2.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.15 + 1.82i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 - 5.70iT - 37T^{2} \)
41 \( 1 + (5.08 + 2.93i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.05 + 4.65i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.70 - 2.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.30 - 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 - 3.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.29 - 7.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.43 + 9.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.83 + 4.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.15 + 8.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.15 + 2.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.63iT - 83T^{2} \)
89 \( 1 + (3.11 - 1.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.05 + 3.49i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.82210628345960113389670697895, −10.19347883787436686676245289404, −9.465517153165459682012441642007, −8.300102618350016070543386945420, −7.929997614769055598587948057347, −6.92065331529347831669913747489, −6.05827943579215218795723019426, −4.97134826258019483976327023414, −2.90383464048925645354545196134, −1.62569127857075840581769174211, 1.06608930393224093603276272067, 3.24647408364644649328634403899, 3.87065867432408172786189837987, 4.71370200172601678488537376230, 6.64481926928904894184308317179, 8.145536366698065402172297303398, 8.432460077283907434250897399912, 9.667057312413603194916697847194, 10.19999226992397407121886952925, 10.95202363532466489682471051132

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