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Properties

Label 2-380-95.94-c2-0-11
Degree $2$
Conductor $380$
Sign $0.675 - 0.737i$
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  + 5.61·3-s + (1.48 + 4.77i)5-s + 7.03i·7-s + 22.5·9-s − 1.02·11-s − 11.3·13-s + (8.36 + 26.7i)15-s − 16.5i·17-s + (−9.54 − 16.4i)19-s + 39.4i·21-s + 35.1i·23-s + (−20.5 + 14.2i)25-s + 75.9·27-s − 6.63i·29-s − 25.3i·31-s + ⋯
L(s)  = 1  + 1.87·3-s + (0.297 + 0.954i)5-s + 1.00i·7-s + 2.50·9-s − 0.0928·11-s − 0.872·13-s + (0.557 + 1.78i)15-s − 0.975i·17-s + (−0.502 − 0.864i)19-s + 1.88i·21-s + 1.52i·23-s + (−0.822 + 0.568i)25-s + 2.81·27-s − 0.228i·29-s − 0.818i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(10.3542\)
Root analytic conductor: \(3.21780\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.95969 + 1.30159i\)
\(L(\frac12)\) \(\approx\) \(2.95969 + 1.30159i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.48 - 4.77i)T \)
19 \( 1 + (9.54 + 16.4i)T \)
good3 \( 1 - 5.61T + 9T^{2} \)
7 \( 1 - 7.03iT - 49T^{2} \)
11 \( 1 + 1.02T + 121T^{2} \)
13 \( 1 + 11.3T + 169T^{2} \)
17 \( 1 + 16.5iT - 289T^{2} \)
23 \( 1 - 35.1iT - 529T^{2} \)
29 \( 1 + 6.63iT - 841T^{2} \)
31 \( 1 + 25.3iT - 961T^{2} \)
37 \( 1 - 47.8T + 1.36e3T^{2} \)
41 \( 1 + 12.1iT - 1.68e3T^{2} \)
43 \( 1 + 50.0iT - 1.84e3T^{2} \)
47 \( 1 + 78.3iT - 2.20e3T^{2} \)
53 \( 1 - 45.0T + 2.80e3T^{2} \)
59 \( 1 - 9.44iT - 3.48e3T^{2} \)
61 \( 1 + 43.1T + 3.72e3T^{2} \)
67 \( 1 - 112.T + 4.48e3T^{2} \)
71 \( 1 + 40.3iT - 5.04e3T^{2} \)
73 \( 1 - 45.4iT - 5.32e3T^{2} \)
79 \( 1 - 144. iT - 6.24e3T^{2} \)
83 \( 1 - 56.8iT - 6.88e3T^{2} \)
89 \( 1 + 81.8iT - 7.92e3T^{2} \)
97 \( 1 + 103.T + 9.40e3T^{2} \)
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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

−11.22031863721337231396727854561, −9.836400359690359958595294706774, −9.497370433431470635341905361804, −8.576100177548875713631821711525, −7.54234314416681586197965512570, −6.92037271363337257851537805596, −5.37402069297525190045861636488, −3.85954281525694702053978866463, −2.65988820191810871636676367082, −2.25850608723946550506574558533, 1.34637246410947585287315476747, 2.58435283599740164008370843067, 3.97755521407661301874725510277, 4.62484023555381161732548959173, 6.45975601993055790161161513521, 7.71481477979642858778985075711, 8.202494558054165932836747012021, 9.088795174629250844131178594002, 9.924512494266176655369644911504, 10.57438659867645874501056622423

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