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Properties

Label 2-380-95.33-c1-0-3
Degree $2$
Conductor $380$
Sign $0.884 - 0.466i$
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  + (0.377 + 0.538i)3-s + (2.02 + 0.943i)5-s + (1.97 + 0.528i)7-s + (0.878 − 2.41i)9-s + (−0.906 + 1.57i)11-s + (−1.44 − 1.01i)13-s + (0.256 + 1.44i)15-s + (0.0577 − 0.123i)17-s + (−2.93 + 3.21i)19-s + (0.459 + 1.26i)21-s + (6.07 + 0.531i)23-s + (3.21 + 3.82i)25-s + (3.53 − 0.947i)27-s + (−3.14 − 1.14i)29-s + (0.680 − 0.393i)31-s + ⋯
L(s)  = 1  + (0.217 + 0.311i)3-s + (0.906 + 0.422i)5-s + (0.745 + 0.199i)7-s + (0.292 − 0.804i)9-s + (−0.273 + 0.473i)11-s + (−0.401 − 0.281i)13-s + (0.0661 + 0.373i)15-s + (0.0140 − 0.0300i)17-s + (−0.674 + 0.738i)19-s + (0.100 + 0.275i)21-s + (1.26 + 0.110i)23-s + (0.643 + 0.765i)25-s + (0.680 − 0.182i)27-s + (−0.583 − 0.212i)29-s + (0.122 − 0.0705i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.884 - 0.466i$
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: \((2,\ 380,\ (\ :1/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71411 + 0.424454i\)
\(L(\frac12)\) \(\approx\) \(1.71411 + 0.424454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 - 0.943i)T \)
19 \( 1 + (2.93 - 3.21i)T \)
good3 \( 1 + (-0.377 - 0.538i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (-1.97 - 0.528i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.906 - 1.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.44 + 1.01i)T + (4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.0577 + 0.123i)T + (-10.9 - 13.0i)T^{2} \)
23 \( 1 + (-6.07 - 0.531i)T + (22.6 + 3.99i)T^{2} \)
29 \( 1 + (3.14 + 1.14i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-0.680 + 0.393i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.83 + 4.83i)T + 37iT^{2} \)
41 \( 1 + (-5.31 - 0.937i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (-0.277 - 3.17i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (5.00 - 2.33i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-0.933 + 10.6i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (8.38 - 3.05i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.98 - 5.01i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.83 + 8.21i)T + (-43.0 + 51.3i)T^{2} \)
71 \( 1 + (1.87 - 2.23i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 + (-2.70 + 1.89i)T + (24.9 - 68.5i)T^{2} \)
79 \( 1 + (2.28 - 12.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-4.13 + 15.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.72 + 9.80i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.79 + 2.70i)T + (62.3 + 74.3i)T^{2} \)
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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

−11.26329508704493935276448948798, −10.41008655915589649900598658967, −9.631785480969638680908156355374, −8.863352809029075530432571309956, −7.66669439065286407445285397793, −6.65426349694523148573141058802, −5.58936190985939270408678665375, −4.54439147250437598551854076206, −3.11899091722174309595213509749, −1.78129186355305109320639253813, 1.50308730525740787995219387579, 2.67089565222915638846557901435, 4.60660784017977622826836484982, 5.26942395387661808225363599793, 6.58574600784916613933938548674, 7.59431166931366555222987744740, 8.547959948751007987470727094595, 9.321882367688725481888630482994, 10.54163847234058782706819877851, 11.04844687143030554619560762633

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