Properties

Label 2-380-380.239-c2-0-64
Degree $2$
Conductor $380$
Sign $0.600 + 0.799i$
Analytic cond. $10.3542$
Root an. cond. $3.21780$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 + 0.809i)2-s + (−0.0580 + 0.100i)3-s + (2.69 − 2.96i)4-s + (2.00 − 4.58i)5-s + (0.0248 − 0.230i)6-s − 7.81·7-s + (−2.52 + 7.59i)8-s + (4.49 + 7.78i)9-s + (0.0430 + 9.99i)10-s − 3.02i·11-s + (0.141 + 0.442i)12-s + (11.0 − 6.40i)13-s + (14.2 − 6.32i)14-s + (0.344 + 0.467i)15-s + (−1.52 − 15.9i)16-s + (17.3 + 10.0i)17-s + ⋯
L(s)  = 1  + (−0.914 + 0.404i)2-s + (−0.0193 + 0.0335i)3-s + (0.672 − 0.740i)4-s + (0.400 − 0.916i)5-s + (0.00413 − 0.0384i)6-s − 1.11·7-s + (−0.315 + 0.948i)8-s + (0.499 + 0.864i)9-s + (0.00430 + 0.999i)10-s − 0.275i·11-s + (0.0117 + 0.0368i)12-s + (0.853 − 0.492i)13-s + (1.02 − 0.451i)14-s + (0.0229 + 0.0311i)15-s + (−0.0952 − 0.995i)16-s + (1.01 + 0.588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.600 + 0.799i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.600 + 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.892719 - 0.446359i\)
\(L(\frac12)\) \(\approx\) \(0.892719 - 0.446359i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.82 - 0.809i)T \)
5 \( 1 + (-2.00 + 4.58i)T \)
19 \( 1 + (-9.66 - 16.3i)T \)
good3 \( 1 + (0.0580 - 0.100i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 + 7.81T + 49T^{2} \)
11 \( 1 + 3.02iT - 121T^{2} \)
13 \( 1 + (-11.0 + 6.40i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (-17.3 - 10.0i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (17.7 + 30.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (20.2 + 35.1i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + 31.8iT - 961T^{2} \)
37 \( 1 + 47.3iT - 1.36e3T^{2} \)
41 \( 1 + (-15.6 + 27.0i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-16.3 + 28.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-19.2 - 33.3i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-62.0 + 35.8i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-85.4 - 49.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-17.0 - 29.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (40.0 + 69.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (14.5 + 8.38i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (57.9 + 33.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (87.2 + 50.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 38.3T + 6.88e3T^{2} \)
89 \( 1 + (22.1 + 38.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-102. - 59.0i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−10.46991415248087164156795221897, −10.10183354771575495809103178198, −9.166439221110188630347308880823, −8.232851075395697479514060465355, −7.51970685472915327842593529064, −5.97135721523897731295425160968, −5.72261837488070706074198556312, −3.96650331516979401726992036750, −2.15128350505529049266325024574, −0.66534779691154773247415998101, 1.31761969686316161771095453584, 3.01266429617561534004717278600, 3.65175357223619153250349324869, 5.86874941301338226965459919388, 6.86987917681579062522397368366, 7.27700348734789975702698156613, 8.832144252842644088699451059517, 9.766112794716220816705498193502, 9.955238551477274033390666673598, 11.21903415906451674493640018531

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