↑

Properties

Label 2-380-76.27-c1-0-17
Degree $2$
Conductor $380$
Sign $0.951 - 0.309i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.33 + 0.474i)2-s + (−0.435 + 0.754i)3-s + (1.55 − 1.26i)4-s + (−0.5 + 0.866i)5-s + (0.222 − 1.21i)6-s − 3.15i·7-s + (−1.46 + 2.41i)8-s + (1.12 + 1.94i)9-s + (0.255 − 1.39i)10-s − 2.82i·11-s + (0.277 + 1.71i)12-s + (3.26 − 1.88i)13-s + (1.49 + 4.20i)14-s + (−0.435 − 0.754i)15-s + (0.806 − 3.91i)16-s + (−0.918 + 1.59i)17-s + ⋯
L(s)  = 1  + (−0.942 + 0.335i)2-s + (−0.251 + 0.435i)3-s + (0.775 − 0.631i)4-s + (−0.223 + 0.387i)5-s + (0.0908 − 0.494i)6-s − 1.19i·7-s + (−0.518 + 0.855i)8-s + (0.373 + 0.647i)9-s + (0.0808 − 0.439i)10-s − 0.852i·11-s + (0.0802 + 0.496i)12-s + (0.905 − 0.522i)13-s + (0.400 + 1.12i)14-s + (−0.112 − 0.194i)15-s + (0.201 − 0.979i)16-s + (−0.222 + 0.385i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.951 - 0.309i$
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: \((2,\ 380,\ (\ :1/2),\ 0.951 - 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.838123 + 0.132778i\)
\(L(\frac12)\) \(\approx\) \(0.838123 + 0.132778i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.474i)T \)
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 \( 1 + (4.06 + 2.34i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.05570344723900036243357771146, −10.36523799197803658973433949027, −9.892723256457135666925419672171, −8.399910413304628267281488197153, −7.84486487817066836852681149491, −6.79182932640301351899747815862, −5.86478479958768326215444427837, −4.50488216754467115887647703699, −3.11464454075272677123192604386, −1.03955045202503895413782411391, 1.20762702378152851393558297694, 2.60778133206627570238996004189, 4.15571461584983880138155983241, 5.78013285557528846702031002635, 6.74911097727778170562222492511, 7.65081589420406210098464058618, 8.824482708620976321083107296387, 9.252769723578472804570703256999, 10.28758340451269596113030415661, 11.53219065037709544726243933648

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