Properties

Label 2-380-20.19-c2-0-72
Degree $2$
Conductor $380$
Sign $-0.207 + 0.978i$
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  + (−1.70 − 1.03i)2-s + 1.20·3-s + (1.84 + 3.54i)4-s + (3.17 − 3.85i)5-s + (−2.05 − 1.25i)6-s + 6.44·7-s + (0.530 − 7.98i)8-s − 7.54·9-s + (−9.43 + 3.29i)10-s − 4.38i·11-s + (2.22 + 4.27i)12-s − 18.5i·13-s + (−11.0 − 6.68i)14-s + (3.82 − 4.64i)15-s + (−9.19 + 13.0i)16-s + 10.0i·17-s + ⋯
L(s)  = 1  + (−0.854 − 0.519i)2-s + 0.401·3-s + (0.461 + 0.887i)4-s + (0.635 − 0.771i)5-s + (−0.343 − 0.208i)6-s + 0.920·7-s + (0.0662 − 0.997i)8-s − 0.838·9-s + (−0.943 + 0.329i)10-s − 0.398i·11-s + (0.185 + 0.356i)12-s − 1.42i·13-s + (−0.786 − 0.477i)14-s + (0.255 − 0.309i)15-s + (−0.574 + 0.818i)16-s + 0.590i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.207 + 0.978i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ -0.207 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.875112 - 1.08067i\)
\(L(\frac12)\) \(\approx\) \(0.875112 - 1.08067i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.70 + 1.03i)T \)
5 \( 1 + (-3.17 + 3.85i)T \)
19 \( 1 + 4.35iT \)
good3 \( 1 - 1.20T + 9T^{2} \)
7 \( 1 - 6.44T + 49T^{2} \)
11 \( 1 + 4.38iT - 121T^{2} \)
13 \( 1 + 18.5iT - 169T^{2} \)
17 \( 1 - 10.0iT - 289T^{2} \)
23 \( 1 - 16.8T + 529T^{2} \)
29 \( 1 - 14.9T + 841T^{2} \)
31 \( 1 - 10.3iT - 961T^{2} \)
37 \( 1 + 32.8iT - 1.36e3T^{2} \)
41 \( 1 - 33.0T + 1.68e3T^{2} \)
43 \( 1 + 77.3T + 1.84e3T^{2} \)
47 \( 1 - 77.0T + 2.20e3T^{2} \)
53 \( 1 + 77.3iT - 2.80e3T^{2} \)
59 \( 1 + 41.4iT - 3.48e3T^{2} \)
61 \( 1 + 5.37T + 3.72e3T^{2} \)
67 \( 1 + 33.9T + 4.48e3T^{2} \)
71 \( 1 - 65.9iT - 5.04e3T^{2} \)
73 \( 1 + 12.9iT - 5.32e3T^{2} \)
79 \( 1 + 49.5iT - 6.24e3T^{2} \)
83 \( 1 + 131.T + 6.88e3T^{2} \)
89 \( 1 - 146.T + 7.92e3T^{2} \)
97 \( 1 - 58.5iT - 9.40e3T^{2} \)
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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

−10.79023157247897814542348638235, −9.961409143714233890621816322856, −8.764968385127640733486346750404, −8.485660173186421081658004392312, −7.60150715580317136551064218340, −6.02749301771061017731042706297, −4.97805002889888735200903484538, −3.36309024200879156880434226479, −2.17503075098283517645057033638, −0.797096533859914230301070089085, 1.67655472513327644858938867372, 2.73150835130767998336650688263, 4.72132120307759499071970881337, 5.87518909889427568949676448820, 6.85111649091539864568308133277, 7.66307157695549528889290512684, 8.746694606915145061334911119201, 9.357059124993882917195192567190, 10.34073490742965315283285890611, 11.27240495424536158655254522245

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