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Properties

Label 2-380-380.259-c1-0-40
Degree $2$
Conductor $380$
Sign $0.213 + 0.977i$
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  + (−1.31 − 0.522i)2-s + (1.49 − 0.861i)3-s + (1.45 + 1.37i)4-s + (2.15 − 0.594i)5-s + (−2.41 + 0.352i)6-s − 2.20·7-s + (−1.19 − 2.56i)8-s + (−0.0139 + 0.0241i)9-s + (−3.14 − 0.345i)10-s − 4.26i·11-s + (3.35 + 0.797i)12-s + (2.52 − 4.37i)13-s + (2.89 + 1.15i)14-s + (2.70 − 2.74i)15-s + (0.226 + 3.99i)16-s + (2.47 − 1.42i)17-s + ⋯
L(s)  = 1  + (−0.929 − 0.369i)2-s + (0.861 − 0.497i)3-s + (0.726 + 0.686i)4-s + (0.964 − 0.265i)5-s + (−0.984 + 0.143i)6-s − 0.831·7-s + (−0.421 − 0.906i)8-s + (−0.00465 + 0.00805i)9-s + (−0.994 − 0.109i)10-s − 1.28i·11-s + (0.968 + 0.230i)12-s + (0.700 − 1.21i)13-s + (0.772 + 0.307i)14-s + (0.698 − 0.708i)15-s + (0.0566 + 0.998i)16-s + (0.599 − 0.345i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.213 + 0.977i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.213 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993304 - 0.800034i\)
\(L(\frac12)\) \(\approx\) \(0.993304 - 0.800034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 + 0.522i)T \)
5 \( 1 + (-2.15 + 0.594i)T \)
19 \( 1 + (3.03 - 3.12i)T \)
good3 \( 1 + (-1.49 + 0.861i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 + 4.26iT - 11T^{2} \)
13 \( 1 + (-2.52 + 4.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.47 + 1.42i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.151 - 0.261i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.15 - 2.97i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.96T + 31T^{2} \)
37 \( 1 + 5.24T + 37T^{2} \)
41 \( 1 + (-8.43 + 4.87i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.03 + 8.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.14 - 5.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.08 - 10.5i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.530 + 0.918i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.60 - 11.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.69 - 2.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.49 - 4.32i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.80 - 2.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.31 + 5.74i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.63T + 83T^{2} \)
89 \( 1 + (-10.0 - 5.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.77 + 8.26i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.70497597430652046079027972254, −10.28149693315963141490120375958, −9.120217675198975702860248429018, −8.530955595985529215396246413827, −7.81758247823855813756367065645, −6.49080639461758801727599699605, −5.69912478836836463609607246177, −3.35773027441627155071871707633, −2.68612777958884056077945432333, −1.14049047552747614827750859022, 1.89785043701337969523280667492, 3.04360417227081798739254081490, 4.65534472303564598591879601491, 6.41611698325799451764455236589, 6.57561913292098791090848204520, 8.100455714414267326700088988606, 9.012422909181168580556708442497, 9.747578863574736145767664760703, 9.994372913468286183054600231364, 11.21925196680570168037435891738

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