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Properties

Label 2-380-380.379-c1-0-7
Degree $2$
Conductor $380$
Sign $-0.958 + 0.284i$
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  + (−0.846 + 1.13i)2-s + 3.11i·3-s + (−0.568 − 1.91i)4-s + 2.23·5-s + (−3.52 − 2.63i)6-s + (2.65 + 0.978i)8-s − 6.67·9-s + (−1.89 + 2.53i)10-s + 2.92i·11-s + (5.96 − 1.76i)12-s − 6.79·13-s + 6.95i·15-s + (−3.35 + 2.17i)16-s + (5.65 − 7.56i)18-s + 4.35i·19-s + (−1.27 − 4.28i)20-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)2-s + 1.79i·3-s + (−0.284 − 0.958i)4-s + 0.999·5-s + (−1.43 − 1.07i)6-s + (0.938 + 0.345i)8-s − 2.22·9-s + (−0.598 + 0.801i)10-s + 0.883i·11-s + (1.72 − 0.510i)12-s − 1.88·13-s + 1.79i·15-s + (−0.838 + 0.544i)16-s + (1.33 − 1.78i)18-s + 0.999i·19-s + (−0.284 − 0.958i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 + 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.958 + 0.284i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.958 + 0.284i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.130328 - 0.898438i\)
\(L(\frac12)\) \(\approx\) \(0.130328 - 0.898438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.846 - 1.13i)T \)
5 \( 1 - 2.23T \)
19 \( 1 - 4.35iT \)
good3 \( 1 - 3.11iT - 3T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 2.92iT - 11T^{2} \)
13 \( 1 + 6.79T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 12.1T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6.04T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5T + 61T^{2} \)
67 \( 1 - 16.0iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 2.99T + 97T^{2} \)
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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

−11.43990059942372313263225500277, −10.28146652724777858200934082720, −9.825868287952442145741257067650, −9.531670338565482589531094982079, −8.402992714415421042553480902685, −7.18584353439507233515447340587, −5.87910262523663040517874781836, −5.09068860839072798462994682033, −4.34253482082024564060782595710, −2.41864101910281138285851983772, 0.72055009389670173550359412257, 2.16908196039831472953955375367, 2.81616788168418079695592099185, 5.07129627607458153286918662497, 6.36262700614961131802146551658, 7.22417859580590290695418525836, 8.039182396579737155324300625774, 9.034684107219229144949905164379, 9.856787626380587243388821647722, 11.08281535224207619552795245786

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