Properties

Label 2-380-95.12-c1-0-7
Degree $2$
Conductor $380$
Sign $-0.122 + 0.992i$
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  + (0.633 − 2.36i)3-s + (2.23 − 0.133i)5-s + (−2 − 2i)7-s + (−2.59 − 1.50i)9-s + 11-s + (2.36 − 0.633i)13-s + (1.09 − 5.36i)15-s + (−0.366 + 1.36i)17-s + (−4.33 − 0.5i)19-s + (−6 + 3.46i)21-s + (4.96 − 0.598i)25-s + (−4.33 + 7.5i)29-s + 1.73i·31-s + (0.633 − 2.36i)33-s + (−4.73 − 4.19i)35-s + ⋯
L(s)  = 1  + (0.366 − 1.36i)3-s + (0.998 − 0.0599i)5-s + (−0.755 − 0.755i)7-s + (−0.866 − 0.500i)9-s + 0.301·11-s + (0.656 − 0.175i)13-s + (0.283 − 1.38i)15-s + (−0.0887 + 0.331i)17-s + (−0.993 − 0.114i)19-s + (−1.30 + 0.755i)21-s + (0.992 − 0.119i)25-s + (−0.804 + 1.39i)29-s + 0.311i·31-s + (0.110 − 0.411i)33-s + (−0.799 − 0.709i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $-0.122 + 0.992i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ -0.122 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09338 - 1.23678i\)
\(L(\frac12)\) \(\approx\) \(1.09338 - 1.23678i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.133i)T \)
19 \( 1 + (4.33 + 0.5i)T \)
good3 \( 1 + (-0.633 + 2.36i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (2 + 2i)T + 7iT^{2} \)
11 \( 1 - T + 11T^{2} \)
13 \( 1 + (-2.36 + 0.633i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.366 - 1.36i)T + (-14.7 - 8.5i)T^{2} \)
23 \( 1 + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.33 - 7.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \)
41 \( 1 + (-3 + 1.73i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.46 + 1.46i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.09 + 1.09i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.09 + 1.90i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.5 - 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.16 - 11.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.5 + 6.06i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-13.6 - 3.66i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-6.06 - 10.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4 - 4i)T - 83iT^{2} \)
89 \( 1 + (7.79 - 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (18.9 + 5.07i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.02098739701082813591149014588, −10.22727733151913851640552101543, −9.122003384373383866096277086521, −8.328475992391639144265035343192, −7.04867479130688639665981883293, −6.61638033894151443048184960652, −5.60995023172593651087637825041, −3.82544727108315522112133042417, −2.40370342196760048394767382510, −1.18283700329449551835460499333, 2.31845141996152509949966139811, 3.50261109951135930513408051197, 4.62251144387401078142235911267, 5.82207365561848000148719741158, 6.50768420863319499381606035000, 8.279291753172397324165713384788, 9.345199422948549280028150008733, 9.485279929628718840464133187197, 10.47816793487332979644585187356, 11.32434771962609315668066954158

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