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Properties

Label 2-380-95.64-c1-0-4
Degree $2$
Conductor $380$
Sign $0.992 - 0.124i$
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  + (−1.74 + 1.00i)3-s + (−1.95 − 1.08i)5-s − 1.34i·7-s + (0.526 − 0.912i)9-s + 5.25·11-s + (2.10 + 1.21i)13-s + (4.50 − 0.0822i)15-s + (−1.17 + 0.679i)17-s + (2.89 − 3.25i)19-s + (1.35 + 2.34i)21-s + (7.05 + 4.07i)23-s + (2.65 + 4.23i)25-s − 3.91i·27-s + (−1.03 + 1.79i)29-s − 0.513·31-s + ⋯
L(s)  = 1  + (−1.00 + 0.581i)3-s + (−0.875 − 0.484i)5-s − 0.507i·7-s + (0.175 − 0.304i)9-s + 1.58·11-s + (0.584 + 0.337i)13-s + (1.16 − 0.0212i)15-s + (−0.285 + 0.164i)17-s + (0.664 − 0.746i)19-s + (0.295 + 0.511i)21-s + (1.47 + 0.849i)23-s + (0.531 + 0.847i)25-s − 0.754i·27-s + (−0.192 + 0.333i)29-s − 0.0921·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899106 + 0.0563908i\)
\(L(\frac12)\) \(\approx\) \(0.899106 + 0.0563908i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.95 + 1.08i)T \)
19 \( 1 + (-2.89 + 3.25i)T \)
good3 \( 1 + (1.74 - 1.00i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 1.34iT - 7T^{2} \)
11 \( 1 - 5.25T + 11T^{2} \)
13 \( 1 + (-2.10 - 1.21i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.17 - 0.679i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-7.05 - 4.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.03 - 1.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.513T + 31T^{2} \)
37 \( 1 + 5.57iT - 37T^{2} \)
41 \( 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-11.0 + 6.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.82 + 1.63i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (10.1 + 5.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.0175 - 0.0304i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.518 + 0.897i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.664 + 0.383i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.68 - 9.84i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.86 + 1.07i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.48 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.20iT - 83T^{2} \)
89 \( 1 + (3.65 - 6.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.721 - 0.416i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.25344720130282063652976717588, −10.90420536882423885482724663883, −9.439927944660633529534608732620, −8.859626112428125202445723692267, −7.46856717480719950137523753400, −6.60588511468281602664363413336, −5.38430700249703087770852142658, −4.41114992012521602393177064466, −3.63351675432247709955212822178, −0.999266177710867797089584562262, 1.06212632974668678391865638699, 3.15019967042056028089860046756, 4.38830834554357136867977614630, 5.80903020374455358777655620120, 6.54086334516579087287576956235, 7.32866751673510685025462448404, 8.524214528812439555757712691320, 9.448328590342305717088542439941, 10.96879629552311339454562033142, 11.28812617464874641927231614396

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