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Properties

Label 2-380-76.31-c1-0-30
Degree $2$
Conductor $380$
Sign $0.664 + 0.747i$
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  + (1.17 + 0.780i)2-s + (−0.980 − 1.69i)3-s + (0.780 + 1.84i)4-s + (−0.5 − 0.866i)5-s + (0.170 − 2.76i)6-s − 1.97i·7-s + (−0.518 + 2.78i)8-s + (−0.421 + 0.730i)9-s + (0.0867 − 1.41i)10-s − 5.82i·11-s + (2.36 − 3.12i)12-s + (4.54 + 2.62i)13-s + (1.53 − 2.32i)14-s + (−0.980 + 1.69i)15-s + (−2.78 + 2.87i)16-s + (−2.29 − 3.97i)17-s + ⋯
L(s)  = 1  + (0.833 + 0.552i)2-s + (−0.565 − 0.980i)3-s + (0.390 + 0.920i)4-s + (−0.223 − 0.387i)5-s + (0.0694 − 1.12i)6-s − 0.745i·7-s + (−0.183 + 0.983i)8-s + (−0.140 + 0.243i)9-s + (0.0274 − 0.446i)10-s − 1.75i·11-s + (0.681 − 0.903i)12-s + (1.26 + 0.727i)13-s + (0.411 − 0.621i)14-s + (−0.253 + 0.438i)15-s + (−0.695 + 0.718i)16-s + (−0.556 − 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.664 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58571 - 0.712171i\)
\(L(\frac12)\) \(\approx\) \(1.58571 - 0.712171i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.780i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-4.33 - 0.434i)T \)
good3 \( 1 + (0.980 + 1.69i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + 1.97iT - 7T^{2} \)
11 \( 1 + 5.82iT - 11T^{2} \)
13 \( 1 + (-4.54 - 2.62i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.29 + 3.97i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.54 + 1.47i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.95 - 2.85i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 - 6.81iT - 37T^{2} \)
41 \( 1 + (-2.48 + 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.44 - 2.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.5 - 6.66i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.3 + 6.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.17 - 8.96i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.62 - 6.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.99 - 5.19i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.0329 + 0.0570i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.29 + 5.69i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.55 + 2.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.28iT - 83T^{2} \)
89 \( 1 + (-10.0 - 5.79i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.927 + 0.535i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.54700213454027755513125517279, −10.85680393256147540547806842551, −9.018120401921948208784908966518, −8.162898296815053677806954517454, −7.18856007646839600390288408756, −6.39674486486170114501783496921, −5.66897884394391618774230908734, −4.32764435421596303344111160011, −3.22077849411161640227285301414, −1.05708941168209296842979157918, 2.06753365270329803775529261475, 3.59489173633837738762870699314, 4.47051896069783083526956717911, 5.44781899806247533449228415441, 6.26867686486559437881226020499, 7.58465774431572583066819008498, 9.123408243169504165346933237614, 10.08003452486147535866196232505, 10.60897599713510069076060234131, 11.44990292615888043387844850015

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