↑

Properties

Label 2-370-185.88-c1-0-16
Degree $2$
Conductor $370$
Sign $-0.651 + 0.758i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (0.5 − 0.866i)2-s + (−0.243 − 0.0652i)3-s + (−0.499 − 0.866i)4-s + (2.10 − 0.752i)5-s + (−0.178 + 0.178i)6-s + (−3.85 − 1.03i)7-s − 0.999·8-s + (−2.54 − 1.46i)9-s + (0.401 − 2.19i)10-s − 3.49i·11-s + (0.0652 + 0.243i)12-s + (−0.330 − 0.572i)13-s + (−2.81 + 2.81i)14-s + (−0.561 + 0.0458i)15-s + (−0.5 + 0.866i)16-s + (5.47 + 3.15i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.140 − 0.0376i)3-s + (−0.249 − 0.433i)4-s + (0.941 − 0.336i)5-s + (−0.0727 + 0.0727i)6-s + (−1.45 − 0.390i)7-s − 0.353·8-s + (−0.847 − 0.489i)9-s + (0.126 − 0.695i)10-s − 1.05i·11-s + (0.0188 + 0.0702i)12-s + (−0.0917 − 0.158i)13-s + (−0.753 + 0.753i)14-s + (−0.145 + 0.0118i)15-s + (−0.125 + 0.216i)16-s + (1.32 + 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.651 + 0.758i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.651 + 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543126 - 1.18275i\)
\(L(\frac12)\) \(\approx\) \(0.543126 - 1.18275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 + (-2.10 + 0.752i)T \)
37 \( 1 + (1.80 - 5.80i)T \)
good3 \( 1 + (0.243 + 0.0652i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.85 + 1.03i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 3.49iT - 11T^{2} \)
13 \( 1 + (0.330 + 0.572i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.47 - 3.15i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.53 + 5.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.13T + 23T^{2} \)
29 \( 1 + (5.40 - 5.40i)T - 29iT^{2} \)
31 \( 1 + (3.66 + 3.66i)T + 31iT^{2} \)
41 \( 1 + (-0.539 + 0.311i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.69T + 43T^{2} \)
47 \( 1 + (-1.94 + 1.94i)T - 47iT^{2} \)
53 \( 1 + (-13.2 + 3.54i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.80 + 0.484i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.86 - 6.96i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.24 - 8.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.54 - 6.13i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.89 + 5.89i)T - 73iT^{2} \)
79 \( 1 + (-3.09 + 11.5i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.181 - 0.0485i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.308 - 1.15i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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

−11.02130346894559940867581550302, −10.20456079115447583093952766317, −9.337296349196896329761342953897, −8.721614128979662864142305137828, −6.97393155077302244151791693773, −5.92784240997534000392165578813, −5.41798278691528907117121728394, −3.59051892525891038306948850110, −2.85119474078972369875958012599, −0.801067398888891890269196231301, 2.44612661796077004535661020609, 3.55756273884250835552019114470, 5.35722766583174052106984817708, 5.79337067848756458208678453583, 6.85629108328978890311508242462, 7.72664781441365619868427481345, 9.252299626134347116723939951336, 9.655173223797436910723299410098, 10.64881422225194136182958372228, 12.07494423774351152668062181748

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