↑

Properties

Label 2-370-185.8-c1-0-11
Degree $2$
Conductor $370$
Sign $0.726 + 0.687i$
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  + (0.866 − 0.5i)2-s + (−0.0652 − 0.243i)3-s + (0.499 − 0.866i)4-s + (1.44 − 1.70i)5-s + (−0.178 − 0.178i)6-s + (1.03 + 3.85i)7-s − 0.999i·8-s + (2.54 − 1.46i)9-s + (0.401 − 2.19i)10-s + 3.49i·11-s + (−0.243 − 0.0652i)12-s + (−0.572 − 0.330i)13-s + (2.81 + 2.81i)14-s + (−0.509 − 0.241i)15-s + (−0.5 − 0.866i)16-s + (−3.15 − 5.47i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.0376 − 0.140i)3-s + (0.249 − 0.433i)4-s + (0.647 − 0.762i)5-s + (−0.0727 − 0.0727i)6-s + (0.390 + 1.45i)7-s − 0.353i·8-s + (0.847 − 0.489i)9-s + (0.126 − 0.695i)10-s + 1.05i·11-s + (−0.0702 − 0.0188i)12-s + (−0.158 − 0.0917i)13-s + (0.753 + 0.753i)14-s + (−0.131 − 0.0622i)15-s + (−0.125 − 0.216i)16-s + (−0.765 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99662 - 0.795407i\)
\(L(\frac12)\) \(\approx\) \(1.99662 - 0.795407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-1.44 + 1.70i)T \)
37 \( 1 + (5.80 - 1.80i)T \)
good3 \( 1 + (0.0652 + 0.243i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.03 - 3.85i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 3.49iT - 11T^{2} \)
13 \( 1 + (0.572 + 0.330i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.15 + 5.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.53 + 5.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 3.13iT - 23T^{2} \)
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 \( 1 - 4.69iT - 43T^{2} \)
47 \( 1 + (-1.94 + 1.94i)T - 47iT^{2} \)
53 \( 1 + (3.54 - 13.2i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (8.38 - 2.24i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-0.0485 + 0.181i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.308 - 1.15i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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

−11.65492804678195762287540872550, −10.38813121108067562867803564721, −9.223026303558770860198398662665, −9.019293794786719049441689085338, −7.29296177252533152592681783306, −6.34499023892226831980205961582, −5.03114134337212721137541589809, −4.71629922037067526859135632365, −2.71504513880552402338364500039, −1.64679202508413657845649538756, 1.89461604279253626753094295568, 3.65700884643540148273550187493, 4.39350152177456422457773057328, 5.82811246314601006948459114846, 6.65201593588182503938400215107, 7.55980613053864608016117591862, 8.464577035855204041912952848560, 10.15495611121378444400228196255, 10.51883452469299211534795876606, 11.28516893224909139294461571125

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