↑

Properties

Label 2-370-37.7-c1-0-13
Degree $2$
Conductor $370$
Sign $0.200 + 0.979i$
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  + (0.939 − 0.342i)2-s + (0.465 + 0.169i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)5-s + 0.495·6-s + (−0.827 − 4.69i)7-s + (0.500 − 0.866i)8-s + (−2.10 − 1.77i)9-s + (−0.5 − 0.866i)10-s + (0.0178 − 0.0309i)11-s + (0.465 − 0.169i)12-s + (−2.22 + 1.86i)13-s + (−2.38 − 4.12i)14-s + (0.0860 − 0.488i)15-s + (0.173 − 0.984i)16-s + (4.81 + 4.03i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.268 + 0.0978i)3-s + (0.383 − 0.321i)4-s + (−0.0776 − 0.440i)5-s + 0.202·6-s + (−0.312 − 1.77i)7-s + (0.176 − 0.306i)8-s + (−0.703 − 0.590i)9-s + (−0.158 − 0.273i)10-s + (0.00538 − 0.00932i)11-s + (0.134 − 0.0489i)12-s + (−0.617 + 0.517i)13-s + (−0.636 − 1.10i)14-s + (0.0222 − 0.126i)15-s + (0.0434 − 0.246i)16-s + (1.16 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.200 + 0.979i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.200 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50490 - 1.22752i\)
\(L(\frac12)\) \(\approx\) \(1.50490 - 1.22752i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (2.58 - 5.50i)T \)
good3 \( 1 + (-0.465 - 0.169i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (0.827 + 4.69i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-0.0178 + 0.0309i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.22 - 1.86i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.81 - 4.03i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-3.55 - 1.29i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3.32 - 5.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.13 + 3.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.81T + 31T^{2} \)
41 \( 1 + (-0.219 + 0.184i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 + (0.0422 + 0.0732i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.17 + 12.3i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (1.71 - 9.73i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.41 + 6.22i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.0868 + 0.492i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-8.73 - 3.17i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 8.43T + 73T^{2} \)
79 \( 1 + (-0.728 - 4.13i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.25 + 4.41i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-1.99 + 11.3i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-2.46 - 4.27i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.43696542658658034740215712074, −10.09554548374302744203102096762, −9.784423116282465467466570038251, −8.292174300951340076000668427694, −7.35243060986559456528759698159, −6.35763816678760397755991138605, −5.11729494209236559257727416634, −3.95231273757835565805735873352, −3.23908187562038083429600444618, −1.14125759963014103787280022754, 2.67515096510939678154234789262, 2.95116711279917862698543724336, 5.05387526855574602965376494932, 5.56551076592850726825269675554, 6.73976568191987678606309859057, 7.83067122964201846060353433039, 8.691985692322895329783750487133, 9.679805128354633092269375571693, 10.87281694631593473032931766656, 11.99915603928681963490512058743

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