Properties

Label 2-380-95.13-c1-0-1
Degree $2$
Conductor $380$
Sign $-0.328 - 0.944i$
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  + (0.224 − 0.104i)3-s + (−2.15 + 0.611i)5-s + (0.357 + 1.33i)7-s + (−1.88 + 2.25i)9-s + (0.00188 + 0.00325i)11-s + (−1.47 + 3.16i)13-s + (−0.418 + 0.362i)15-s + (0.0221 − 0.253i)17-s + (0.384 + 4.34i)19-s + (0.219 + 0.262i)21-s + (−3.63 + 2.54i)23-s + (4.25 − 2.62i)25-s + (−0.380 + 1.42i)27-s + (1.17 + 0.989i)29-s + (0.377 + 0.218i)31-s + ⋯
L(s)  = 1  + (0.129 − 0.0604i)3-s + (−0.961 + 0.273i)5-s + (0.135 + 0.504i)7-s + (−0.629 + 0.750i)9-s + (0.000566 + 0.000981i)11-s + (−0.409 + 0.877i)13-s + (−0.108 + 0.0935i)15-s + (0.00537 − 0.0614i)17-s + (0.0882 + 0.996i)19-s + (0.0479 + 0.0571i)21-s + (−0.757 + 0.530i)23-s + (0.850 − 0.525i)25-s + (−0.0732 + 0.273i)27-s + (0.218 + 0.183i)29-s + (0.0678 + 0.0391i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.328 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489025 + 0.687812i\)
\(L(\frac12)\) \(\approx\) \(0.489025 + 0.687812i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.611i)T \)
19 \( 1 + (-0.384 - 4.34i)T \)
good3 \( 1 + (-0.224 + 0.104i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-0.357 - 1.33i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.00188 - 0.00325i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.47 - 3.16i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (-0.0221 + 0.253i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (3.63 - 2.54i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-1.17 - 0.989i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.377 - 0.218i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.496 - 0.496i)T + 37iT^{2} \)
41 \( 1 + (-2.34 + 6.44i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (3.96 - 5.66i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (2.70 - 0.236i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (3.00 + 4.28i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-7.08 + 5.94i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.899 - 5.10i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.01 - 11.5i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (1.85 - 0.327i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (4.92 + 10.5i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (-10.2 - 3.74i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.18 - 0.316i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-4.35 + 1.58i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-17.5 - 1.53i)T + (95.5 + 16.8i)T^{2} \)
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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

−11.72713826519829213346086418470, −10.88014969823601720429705740314, −9.828236557248371608888056829002, −8.655923954838609538279674679117, −7.984377425474576628056520819129, −7.08508473906127451728476360686, −5.83514114050600512778600410418, −4.69368382729869031548671142531, −3.49634628246347793132760799176, −2.15200440665205653350602005886, 0.54659226127604789410489489460, 2.88812184956324696137860742299, 3.98136963636274331732898472811, 5.03896246025570821694164992106, 6.36484128563472205208957251192, 7.47343720745969108728044363289, 8.252870851339572669324619399153, 9.123631524244999343035474141897, 10.22916598042335808429486138416, 11.17438812492467828867722689341

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