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Properties

Label 2-380-95.33-c1-0-8
Degree $2$
Conductor $380$
Sign $-0.919 - 0.393i$
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  + (−0.886 − 1.26i)3-s + (−2.22 − 0.187i)5-s + (0.477 + 0.127i)7-s + (0.209 − 0.575i)9-s + (−1.30 + 2.26i)11-s + (−3.24 − 2.27i)13-s + (1.73 + 2.98i)15-s + (−3.14 + 6.74i)17-s + (−3.99 − 1.74i)19-s + (−0.261 − 0.717i)21-s + (−6.95 − 0.608i)23-s + (4.92 + 0.836i)25-s + (−5.39 + 1.44i)27-s + (7.14 + 2.59i)29-s + (−4.18 + 2.41i)31-s + ⋯
L(s)  = 1  + (−0.511 − 0.730i)3-s + (−0.996 − 0.0839i)5-s + (0.180 + 0.0483i)7-s + (0.0698 − 0.191i)9-s + (−0.394 + 0.683i)11-s + (−0.900 − 0.630i)13-s + (0.448 + 0.771i)15-s + (−0.763 + 1.63i)17-s + (−0.916 − 0.399i)19-s + (−0.0569 − 0.156i)21-s + (−1.45 − 0.126i)23-s + (0.985 + 0.167i)25-s + (−1.03 + 0.278i)27-s + (1.32 + 0.482i)29-s + (−0.751 + 0.433i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.919 - 0.393i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.919 - 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0219298 + 0.106873i\)
\(L(\frac12)\) \(\approx\) \(0.0219298 + 0.106873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 0.187i)T \)
19 \( 1 + (3.99 + 1.74i)T \)
good3 \( 1 + (0.886 + 1.26i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (-0.477 - 0.127i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.30 - 2.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.24 + 2.27i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (3.14 - 6.74i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (6.95 + 0.608i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (-7.14 - 2.59i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (4.18 - 2.41i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.46 + 2.46i)T + 37iT^{2} \)
41 \( 1 + (-7.64 - 1.34i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.596 + 6.82i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-3.78 + 1.76i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-1.10 + 12.6i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (9.64 - 3.51i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-4.23 + 3.55i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (2.96 + 6.36i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (-2.25 + 2.68i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (7.90 - 5.53i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (-2.00 + 11.3i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.901 - 3.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-1.27 - 7.22i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-1.27 - 0.594i)T + (62.3 + 74.3i)T^{2} \)
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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

−10.89584610692622178937550571652, −10.18847696440571360439873626513, −8.740207088696520412439013614639, −7.918654177144047499926361521664, −7.07467946557968885664172406635, −6.19893543645779167734529110131, −4.86413461939468319013195281459, −3.82506398559720136642032839117, −2.04628432543922537740900140825, −0.07324483910449743436519030623, 2.61816454245944547742800456885, 4.24570796135482700120923239833, 4.71251932686791044996435213128, 6.03401070059519636749931225648, 7.31139675495532305205595458432, 8.075466957446226897638941512702, 9.223643462500288396394561216345, 10.20631776512737915587941170947, 11.05854340342651773611260466128, 11.63071409782110111461324371508

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