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Properties

Label 2-380-76.27-c1-0-10
Degree $2$
Conductor $380$
Sign $0.999 + 0.00624i$
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  + (−0.255 − 1.39i)2-s + (0.435 − 0.754i)3-s + (−1.86 + 0.710i)4-s + (−0.5 + 0.866i)5-s + (−1.16 − 0.412i)6-s + 3.15i·7-s + (1.46 + 2.41i)8-s + (1.12 + 1.94i)9-s + (1.33 + 0.474i)10-s + 2.82i·11-s + (−0.277 + 1.71i)12-s + (3.26 − 1.88i)13-s + (4.39 − 0.806i)14-s + (0.435 + 0.754i)15-s + (2.98 − 2.65i)16-s + (−0.918 + 1.59i)17-s + ⋯
L(s)  = 1  + (−0.180 − 0.983i)2-s + (0.251 − 0.435i)3-s + (−0.934 + 0.355i)4-s + (−0.223 + 0.387i)5-s + (−0.473 − 0.168i)6-s + 1.19i·7-s + (0.518 + 0.855i)8-s + (0.373 + 0.647i)9-s + (0.421 + 0.149i)10-s + 0.852i·11-s + (−0.0802 + 0.496i)12-s + (0.905 − 0.522i)13-s + (1.17 − 0.215i)14-s + (0.112 + 0.194i)15-s + (0.747 − 0.664i)16-s + (−0.222 + 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.999 + 0.00624i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.999 + 0.00624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14205 - 0.00356876i\)
\(L(\frac12)\) \(\approx\) \(1.14205 - 0.00356876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.255 + 1.39i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (4.35 - 0.244i)T \)
good3 \( 1 + (-0.435 + 0.754i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.15iT - 7T^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (-3.26 + 1.88i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.918 - 1.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.954 + 0.551i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.92 + 2.84i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 + (-5.31 - 3.07i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.71 - 0.989i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.537 - 0.310i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.4 + 7.16i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.49 + 6.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.91 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.58 - 9.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.66 + 8.07i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0750 - 0.130i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.12 + 1.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (7.77 - 4.48i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.06 + 2.34i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.28138174748062008153154141103, −10.58270482554608415186658891982, −9.638032302151582372673736203541, −8.520000798716141786958788733837, −8.036882900608414666403251519651, −6.69199983502243151757218457951, −5.33648520638305944942226112559, −4.13552690197602114012019717316, −2.73483526402250857857798102907, −1.81569462219192749792650041480, 0.866552046185560672115451773050, 3.80881694179186904186171385951, 4.21475467074394695244086809538, 5.65889130879953079162694039577, 6.75527976987208750541010149845, 7.50492372583101201584459930918, 8.850442681762614642797210823928, 9.028032087587741353262266298731, 10.39909021314053295538186134750, 10.98388422585799267309843335518

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