Properties

Label 2-380-19.11-c1-0-1
Degree $2$
Conductor $380$
Sign $0.550 - 0.834i$
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  + (0.176 − 0.306i)3-s + (−0.5 + 0.866i)5-s − 4.30·7-s + (1.43 + 2.48i)9-s + 6.01·11-s + (2.97 + 5.15i)13-s + (0.176 + 0.306i)15-s + (−1.93 + 3.35i)17-s + (4.19 + 1.17i)19-s + (−0.760 + 1.31i)21-s + (−0.391 − 0.678i)23-s + (−0.499 − 0.866i)25-s + 2.07·27-s + (−3.98 − 6.89i)29-s − 4.49·31-s + ⋯
L(s)  = 1  + (0.102 − 0.176i)3-s + (−0.223 + 0.387i)5-s − 1.62·7-s + (0.479 + 0.829i)9-s + 1.81·11-s + (0.825 + 1.42i)13-s + (0.0456 + 0.0790i)15-s + (−0.469 + 0.813i)17-s + (0.963 + 0.268i)19-s + (−0.166 + 0.287i)21-s + (−0.0816 − 0.141i)23-s + (−0.0999 − 0.173i)25-s + 0.399·27-s + (−0.739 − 1.28i)29-s − 0.806·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.550 - 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08584 + 0.584624i\)
\(L(\frac12)\) \(\approx\) \(1.08584 + 0.584624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-4.19 - 1.17i)T \)
good3 \( 1 + (-0.176 + 0.306i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 4.30T + 7T^{2} \)
11 \( 1 - 6.01T + 11T^{2} \)
13 \( 1 + (-2.97 - 5.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.93 - 3.35i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.391 + 0.678i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.98 + 6.89i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.49T + 31T^{2} \)
37 \( 1 + 0.988T + 37T^{2} \)
41 \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.785 + 1.35i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.630 - 1.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.07 - 7.05i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.62 - 4.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.80 + 4.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.52 + 6.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.90 + 5.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.62 + 8.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.99 + 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.58T + 83T^{2} \)
89 \( 1 + (-1.69 - 2.93i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.69 + 6.40i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.56843150247669070854098046431, −10.58718483135170451349348043548, −9.486480081310959135195807102375, −9.030811402634360671318517252239, −7.58815187123619367005842546303, −6.58471269946863474281890460769, −6.18991050771967944697574787868, −4.20202114297450432123922315205, −3.51350988179190147559305669065, −1.77276247746210895030994729784, 0.900455901628262439171570114433, 3.35389435248843869260378582265, 3.77608113204582642130537463465, 5.47097535062547958703877859134, 6.55203189584622777505682281824, 7.19642065431515107515200970674, 8.852544826887631995466728244540, 9.285392356272437960176423095557, 10.06154324752920913544187943111, 11.29518569964965617020215364585

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