↑

Properties

Label 2-380-95.13-c1-0-3
Degree $2$
Conductor $380$
Sign $0.875 - 0.483i$
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  + (−0.362 + 0.169i)3-s + (1.57 − 1.58i)5-s + (1.00 + 3.76i)7-s + (−1.82 + 2.17i)9-s + (0.973 + 1.68i)11-s + (2.46 − 5.27i)13-s + (−0.303 + 0.841i)15-s + (−0.350 + 4.00i)17-s + (4.22 + 1.08i)19-s + (−1.00 − 1.19i)21-s + (1.79 − 1.25i)23-s + (−0.0286 − 4.99i)25-s + (0.604 − 2.25i)27-s + (3.44 + 2.89i)29-s + (2.99 + 1.72i)31-s + ⋯
L(s)  = 1  + (−0.209 + 0.0976i)3-s + (0.705 − 0.709i)5-s + (0.380 + 1.42i)7-s + (−0.608 + 0.725i)9-s + (0.293 + 0.508i)11-s + (0.682 − 1.46i)13-s + (−0.0783 + 0.217i)15-s + (−0.0850 + 0.971i)17-s + (0.968 + 0.249i)19-s + (−0.218 − 0.260i)21-s + (0.374 − 0.262i)23-s + (−0.00573 − 0.999i)25-s + (0.116 − 0.434i)27-s + (0.640 + 0.537i)29-s + (0.537 + 0.310i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 - 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(380\)    =    \(2^{2} \cdot 5 \cdot 19\)
Sign: $0.875 - 0.483i$
Analytic conductor: \(3.03431\)
Root analytic conductor: \(1.74192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{380} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 380,\ (\ :1/2),\ 0.875 - 0.483i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42155 + 0.366765i\)
\(L(\frac12)\) \(\approx\) \(1.42155 + 0.366765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 + 1.58i)T \)
19 \( 1 + (-4.22 - 1.08i)T \)
good3 \( 1 + (0.362 - 0.169i)T + (1.92 - 2.29i)T^{2} \)
7 \( 1 + (-1.00 - 3.76i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.973 - 1.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.46 + 5.27i)T + (-8.35 - 9.95i)T^{2} \)
17 \( 1 + (0.350 - 4.00i)T + (-16.7 - 2.95i)T^{2} \)
23 \( 1 + (-1.79 + 1.25i)T + (7.86 - 21.6i)T^{2} \)
29 \( 1 + (-3.44 - 2.89i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-2.99 - 1.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.01 + 5.01i)T + 37iT^{2} \)
41 \( 1 + (3.17 - 8.73i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.28 - 1.82i)T + (-14.7 - 40.4i)T^{2} \)
47 \( 1 + (4.06 - 0.355i)T + (46.2 - 8.16i)T^{2} \)
53 \( 1 + (4.39 + 6.27i)T + (-18.1 + 49.8i)T^{2} \)
59 \( 1 + (-4.29 + 3.60i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-1.94 + 11.0i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.637 + 7.28i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (-2.69 + 0.475i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (1.74 + 3.74i)T + (-46.9 + 55.9i)T^{2} \)
79 \( 1 + (8.14 + 2.96i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (14.3 - 3.85i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.72 - 0.627i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (0.221 + 0.0194i)T + (95.5 + 16.8i)T^{2} \)
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

−11.46342949873009349055560800282, −10.49804726644344340751321138635, −9.585592067130847276701605662998, −8.455644140132631728850950171177, −8.191553525873543098460240614748, −6.32389639450712583075620859228, −5.46868112213000834390539500656, −4.95335400090061133464014123922, −3.01657412210162635069199219153, −1.67629079492611180511977522632, 1.21032161747712779159863294390, 3.06312029593956750436248395628, 4.19238432113941316187602399514, 5.59469696149738328265173086968, 6.74244459059290691982860977675, 7.11733354025267157912279419768, 8.642222439698922882585020635403, 9.530844069498426845682604170478, 10.42237068671274790574304995637, 11.47272042352025884558228051278

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