Properties

Label 2-380-380.159-c2-0-54
Degree $2$
Conductor $380$
Sign $0.0606 - 0.998i$
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  + (1.97 + 0.339i)2-s + (1.97 + 3.42i)3-s + (3.76 + 1.33i)4-s + (−3.94 − 3.07i)5-s + (2.73 + 7.41i)6-s + 2.56·7-s + (6.97 + 3.91i)8-s + (−3.30 + 5.73i)9-s + (−6.72 − 7.40i)10-s + 17.7i·11-s + (2.87 + 15.5i)12-s + (4.58 + 2.64i)13-s + (5.06 + 0.871i)14-s + (2.73 − 19.5i)15-s + (12.4 + 10.0i)16-s + (−1.90 + 1.09i)17-s + ⋯
L(s)  = 1  + (0.985 + 0.169i)2-s + (0.658 + 1.14i)3-s + (0.942 + 0.334i)4-s + (−0.788 − 0.615i)5-s + (0.455 + 1.23i)6-s + 0.367·7-s + (0.872 + 0.489i)8-s + (−0.367 + 0.636i)9-s + (−0.672 − 0.740i)10-s + 1.61i·11-s + (0.239 + 1.29i)12-s + (0.352 + 0.203i)13-s + (0.361 + 0.0622i)14-s + (0.182 − 1.30i)15-s + (0.776 + 0.630i)16-s + (−0.111 + 0.0645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0606 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0606 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.0606 - 0.998i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.0606 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.60377 + 2.45046i\)
\(L(\frac12)\) \(\approx\) \(2.60377 + 2.45046i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.97 - 0.339i)T \)
5 \( 1 + (3.94 + 3.07i)T \)
19 \( 1 + (15.3 - 11.2i)T \)
good3 \( 1 + (-1.97 - 3.42i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 - 2.56T + 49T^{2} \)
11 \( 1 - 17.7iT - 121T^{2} \)
13 \( 1 + (-4.58 - 2.64i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + (1.90 - 1.09i)T + (144.5 - 250. i)T^{2} \)
23 \( 1 + (-8.21 + 14.2i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-24.5 + 42.4i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + 13.0iT - 961T^{2} \)
37 \( 1 + 53.5iT - 1.36e3T^{2} \)
41 \( 1 + (-12.6 - 21.9i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (36.7 + 63.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (27.9 - 48.4i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (38.6 + 22.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-58.4 + 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (2.07 - 3.59i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-51.0 + 88.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + (-35.9 + 20.7i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (41.1 - 23.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (63.0 - 36.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 124.T + 6.88e3T^{2} \)
89 \( 1 + (-50.2 + 86.9i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-53.4 + 30.8i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−11.47227258757314082976322980819, −10.50683400062084370878219946237, −9.580845552136889696280590790690, −8.459682134487075551516020855552, −7.72761772286977223671885364277, −6.51270524197121913028980198512, −4.93143290712893883468151680378, −4.39356000776742922816913627221, −3.68748758994343630827405141146, −2.13879572265607335769999274623, 1.21533909024577810688906285776, 2.78862410399013823534769907885, 3.44883890792535664856079998327, 4.93598560084704821428450175560, 6.37113698616582106097300635176, 6.93757539765256984865767217281, 8.036048292090410894737457815368, 8.583211217021888949687353645506, 10.49306053798257870515217440581, 11.21860715076300150858558508252

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