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Properties

Label 2-380-19.11-c1-0-7
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  + (1.60 − 2.77i)3-s + (0.5 − 0.866i)5-s − 2.20·7-s + (−3.63 − 6.29i)9-s − 1.20·11-s + (0.5 + 0.866i)13-s + (−1.60 − 2.77i)15-s + (1.07 − 1.85i)17-s + (4.30 + 0.673i)19-s + (−3.53 + 6.11i)21-s + (4.63 + 8.02i)23-s + (−0.499 − 0.866i)25-s − 13.6·27-s + (−4.16 − 7.21i)29-s + 8.26·31-s + ⋯
L(s)  = 1  + (0.925 − 1.60i)3-s + (0.223 − 0.387i)5-s − 0.833·7-s + (−1.21 − 2.09i)9-s − 0.363·11-s + (0.138 + 0.240i)13-s + (−0.413 − 0.716i)15-s + (0.259 − 0.449i)17-s + (0.988 + 0.154i)19-s + (−0.770 + 1.33i)21-s + (0.966 + 1.67i)23-s + (−0.0999 − 0.173i)25-s − 2.63·27-s + (−0.773 − 1.33i)29-s + 1.48·31-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: $\chi_{380} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.869758 - 1.41069i\)
\(L(\frac12)\) \(\approx\) \(0.869758 - 1.41069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-4.30 - 0.673i)T \)
good3 \( 1 + (-1.60 + 2.77i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.07 + 1.85i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.63 - 8.02i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.16 + 7.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.26T + 31T^{2} \)
37 \( 1 + 2.20T + 37T^{2} \)
41 \( 1 + (-3.30 + 5.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.17 - 2.03i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.16 - 7.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.134 + 0.232i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.16 + 7.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.70 - 2.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.06 - 1.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.23 + 9.06i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.42 + 4.20i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.20 - 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.73T + 83T^{2} \)
89 \( 1 + (-5.40 - 9.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.87 - 13.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42487145989413300757103123019, −9.678834946621356003729392534733, −9.236337730451119378215279763545, −8.058968891241365567146412800265, −7.41430312912326371254087119438, −6.49710013436724130277498090482, −5.49595126821813442174649274860, −3.52149807909189251944083167807, −2.51902408342793371346145184603, −1.07924531106351121287780469930, 2.74807458704823105042353756459, 3.37370794355647503867387934307, 4.59576026771695193083575584189, 5.62524471257861957181020672391, 6.99417085650886463469837316460, 8.305450475758608049545562777778, 9.017889780210457777692520218451, 9.967289694693002386220377768231, 10.36449601531570899340199572462, 11.25487098304225809373457144207

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