Properties

Label 2-380-19.17-c1-0-2
Degree $2$
Conductor $380$
Sign $0.232 - 0.972i$
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  + (1.75 + 1.47i)3-s + (−0.939 + 0.342i)5-s + (1.54 + 2.67i)7-s + (0.387 + 2.19i)9-s + (−0.285 + 0.493i)11-s + (−1.79 + 1.51i)13-s + (−2.14 − 0.782i)15-s + (0.907 − 5.14i)17-s + (−2.34 + 3.67i)19-s + (−1.22 + 6.95i)21-s + (8.47 + 3.08i)23-s + (0.766 − 0.642i)25-s + (0.876 − 1.51i)27-s + (−0.559 − 3.17i)29-s + (−1.68 − 2.91i)31-s + ⋯
L(s)  = 1  + (1.01 + 0.849i)3-s + (−0.420 + 0.152i)5-s + (0.583 + 1.01i)7-s + (0.129 + 0.733i)9-s + (−0.0859 + 0.148i)11-s + (−0.499 + 0.418i)13-s + (−0.555 − 0.202i)15-s + (0.220 − 1.24i)17-s + (−0.537 + 0.843i)19-s + (−0.267 + 1.51i)21-s + (1.76 + 0.642i)23-s + (0.153 − 0.128i)25-s + (0.168 − 0.292i)27-s + (−0.103 − 0.589i)29-s + (−0.302 − 0.523i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.232 - 0.972i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.232 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39547 + 1.10156i\)
\(L(\frac12)\) \(\approx\) \(1.39547 + 1.10156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.939 - 0.342i)T \)
19 \( 1 + (2.34 - 3.67i)T \)
good3 \( 1 + (-1.75 - 1.47i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (-1.54 - 2.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.285 - 0.493i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.79 - 1.51i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.907 + 5.14i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-8.47 - 3.08i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.559 + 3.17i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.68 + 2.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 + (-3.75 - 3.14i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.26 + 0.822i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.782 + 4.43i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (-12.4 - 4.54i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-0.866 + 4.91i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-1.59 - 0.580i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.49 + 14.1i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.26 - 0.825i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (6.11 + 5.12i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-5.59 - 4.69i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (4.06 + 7.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.72 + 3.12i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-1.72 + 9.76i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.59667767958766921393061451437, −10.52490299607123599074792746276, −9.445172415462973665840124484534, −8.960066367197350070391442915200, −8.038596205185725771322883143550, −7.07057526541233561845329456294, −5.46227735990201125819143080414, −4.53254081669783933347950478419, −3.33868635616215448786457716512, −2.29266454530154745020323890118, 1.22040151997546246337850177570, 2.71952908166086898882401784720, 3.95763036358874488854131009882, 5.16651303353301750345453040046, 6.92619923687544342740209722302, 7.37527203869368166832335741437, 8.393608500765904407072108444979, 8.879476375997525232786957000881, 10.47384231701220745236858418050, 10.97544220031288411725563327008

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