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Properties

Label 2-380-19.6-c1-0-4
Degree $2$
Conductor $380$
Sign $0.835 - 0.549i$
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  + (2.92 + 1.06i)3-s + (0.173 + 0.984i)5-s + (0.643 − 1.11i)7-s + (5.13 + 4.31i)9-s + (−2.25 − 3.89i)11-s + (−0.115 + 0.0418i)13-s + (−0.541 + 3.06i)15-s + (−0.730 + 0.613i)17-s + (−1.18 + 4.19i)19-s + (3.07 − 2.57i)21-s + (0.507 − 2.88i)23-s + (−0.939 + 0.342i)25-s + (5.77 + 10.0i)27-s + (−5.12 − 4.30i)29-s + (−2.91 + 5.04i)31-s + ⋯
L(s)  = 1  + (1.69 + 0.615i)3-s + (0.0776 + 0.440i)5-s + (0.243 − 0.421i)7-s + (1.71 + 1.43i)9-s + (−0.678 − 1.17i)11-s + (−0.0319 + 0.0116i)13-s + (−0.139 + 0.792i)15-s + (−0.177 + 0.148i)17-s + (−0.272 + 0.962i)19-s + (0.670 − 0.562i)21-s + (0.105 − 0.600i)23-s + (−0.187 + 0.0684i)25-s + (1.11 + 1.92i)27-s + (−0.952 − 0.799i)29-s + (−0.522 + 0.905i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 - 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.835 - 0.549i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.835 - 0.549i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18394 + 0.653113i\)
\(L(\frac12)\) \(\approx\) \(2.18394 + 0.653113i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (1.18 - 4.19i)T \)
good3 \( 1 + (-2.92 - 1.06i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (-0.643 + 1.11i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.25 + 3.89i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.115 - 0.0418i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.730 - 0.613i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.507 + 2.88i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.12 + 4.30i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (2.91 - 5.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.95T + 37T^{2} \)
41 \( 1 + (-7.45 - 2.71i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.65 + 9.38i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (0.709 + 0.595i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (0.00414 - 0.0234i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-8.19 + 6.87i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.02 + 5.81i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-12.0 - 10.1i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.62 + 14.8i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-3.61 - 1.31i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.80 + 1.01i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (4.36 - 7.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (14.0 - 5.11i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (13.3 - 11.1i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.99839851556899008986597371941, −10.46347690880637091424833364509, −9.565293464745681951512997314665, −8.530075168435559578100703886084, −8.050288061836315446013268505189, −7.01024259509241072148198924461, −5.48540867633390881976388835007, −4.05000788770327655606311795659, −3.29528216479371635990322573736, −2.13154991309907048355371829764, 1.79644525367869500679002965322, 2.68152783562314864104926926031, 4.06986295917648742501198153087, 5.30460010235076807124693886736, 7.00181814404459699022219569786, 7.57738577601237872516340798435, 8.566390408333744364445706149123, 9.205107302929939334673555328851, 9.972880551104570167753428514893, 11.38679778232301242771645776509

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