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Properties

Label 2-380-76.27-c1-0-7
Degree $2$
Conductor $380$
Sign $0.364 - 0.931i$
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  + (−1.41 + 0.0968i)2-s + (0.717 − 1.24i)3-s + (1.98 − 0.273i)4-s + (−0.5 + 0.866i)5-s + (−0.891 + 1.82i)6-s + 2.27i·7-s + (−2.76 + 0.577i)8-s + (0.470 + 0.814i)9-s + (0.621 − 1.27i)10-s + 3.14i·11-s + (1.08 − 2.65i)12-s + (−3.81 + 2.20i)13-s + (−0.220 − 3.21i)14-s + (0.717 + 1.24i)15-s + (3.85 − 1.08i)16-s + (−3.77 + 6.54i)17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0684i)2-s + (0.414 − 0.717i)3-s + (0.990 − 0.136i)4-s + (−0.223 + 0.387i)5-s + (−0.364 + 0.744i)6-s + 0.860i·7-s + (−0.978 + 0.204i)8-s + (0.156 + 0.271i)9-s + (0.196 − 0.401i)10-s + 0.947i·11-s + (0.312 − 0.767i)12-s + (−1.05 + 0.610i)13-s + (−0.0589 − 0.858i)14-s + (0.185 + 0.320i)15-s + (0.962 − 0.270i)16-s + (−0.916 + 1.58i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.364 - 0.931i$
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: \((2,\ 380,\ (\ :1/2),\ 0.364 - 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679852 + 0.464102i\)
\(L(\frac12)\) \(\approx\) \(0.679852 + 0.464102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0968i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.366 + 4.34i)T \)
good3 \( 1 + (-0.717 + 1.24i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 2.27iT - 7T^{2} \)
11 \( 1 - 3.14iT - 11T^{2} \)
13 \( 1 + (3.81 - 2.20i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.77 - 6.54i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.92 + 2.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.79 - 2.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.49T + 31T^{2} \)
37 \( 1 + 7.02iT - 37T^{2} \)
41 \( 1 + (-5.57 - 3.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.32 - 4.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.71 - 4.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.50 - 3.17i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.92 + 5.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.70 - 9.88i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.63 + 6.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.87 + 6.70i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.63 + 11.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.81 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + (-2.24 + 1.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.67 + 0.967i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34516941495970728076747464838, −10.61082006361226285592331163250, −9.452807021890009733725299803284, −8.793685862897153277011878722918, −7.74182020397795623165762268524, −7.09607478711599951765098385799, −6.26299125077572247922008891588, −4.66392924525831495662161463312, −2.62140166078694421229341740798, −1.92938045447741347647793812602, 0.71090045889630236330507513933, 2.82683256328028751770958292185, 3.91285734454006819515280779487, 5.26983264725496282250298919038, 6.79036005973427114795196409593, 7.61964309121559145914059358119, 8.567225166459651700799836062753, 9.477406112884766385118146673764, 9.994453730588507496249409556892, 10.98422020023523971036458049019

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