Properties

Label 2-380-380.239-c2-0-48
Degree $2$
Conductor $380$
Sign $-0.872 - 0.488i$
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  + (1.42 + 1.40i)2-s + (−1.00 + 1.74i)3-s + (0.0597 + 3.99i)4-s + (3.69 + 3.36i)5-s + (−3.89 + 1.07i)6-s + 3.07·7-s + (−5.52 + 5.78i)8-s + (2.46 + 4.26i)9-s + (0.536 + 9.98i)10-s + 3.77i·11-s + (−7.05 − 3.93i)12-s + (9.11 − 5.26i)13-s + (4.38 + 4.31i)14-s + (−9.62 + 3.06i)15-s + (−15.9 + 0.477i)16-s + (−9.93 − 5.73i)17-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s + (−0.336 + 0.582i)3-s + (0.0149 + 0.999i)4-s + (0.739 + 0.673i)5-s + (−0.648 + 0.179i)6-s + 0.439·7-s + (−0.691 + 0.722i)8-s + (0.273 + 0.473i)9-s + (0.0536 + 0.998i)10-s + 0.342i·11-s + (−0.587 − 0.327i)12-s + (0.701 − 0.404i)13-s + (0.312 + 0.308i)14-s + (−0.641 + 0.204i)15-s + (−0.999 + 0.0298i)16-s + (−0.584 − 0.337i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.872 - 0.488i$
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: \((2,\ 380,\ (\ :1),\ -0.872 - 0.488i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.645706 + 2.47331i\)
\(L(\frac12)\) \(\approx\) \(0.645706 + 2.47331i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.42 - 1.40i)T \)
5 \( 1 + (-3.69 - 3.36i)T \)
19 \( 1 + (-16.4 + 9.51i)T \)
good3 \( 1 + (1.00 - 1.74i)T + (-4.5 - 7.79i)T^{2} \)
7 \( 1 - 3.07T + 49T^{2} \)
11 \( 1 - 3.77iT - 121T^{2} \)
13 \( 1 + (-9.11 + 5.26i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + (9.93 + 5.73i)T + (144.5 + 250. i)T^{2} \)
23 \( 1 + (6.96 + 12.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (9.09 + 15.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 - 50.1iT - 961T^{2} \)
37 \( 1 + 17.0iT - 1.36e3T^{2} \)
41 \( 1 + (-35.7 + 61.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (2.56 - 4.44i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (27.8 + 48.1i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (21.0 - 12.1i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-79.5 - 45.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (2.51 + 4.36i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (16.9 + 29.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (-112. - 65.1i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-69.2 - 39.9i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-6.25 - 3.61i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 61.2T + 6.88e3T^{2} \)
89 \( 1 + (-14.2 - 24.6i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (19.9 + 11.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.36799181139623002816938717816, −10.77955324399788575926391450878, −9.758420512523073478579132724001, −8.667158158350977467571916492886, −7.50025035922240909622088002481, −6.68154017494614687784670222975, −5.55747278885791658040611298743, −4.89527347856229398219399557471, −3.67154699598797391453291333586, −2.27519671705630061634513639118, 1.01748933989140341682216921970, 1.92394845547147475876423740809, 3.65296639970858863470262870644, 4.80808316786877052857457796250, 5.91202372283235991503390603848, 6.44721320844340642451381544379, 7.967946346480906276766488347278, 9.258190975216271174680371138749, 9.827947459596414015009600515619, 11.16911023456589924781933609271

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