↑

Properties

Label 2-308-7.4-c3-0-16
Degree $2$
Conductor $308$
Sign $-0.566 + 0.824i$
Analytic cond. $18.1725$
Root an. cond. $4.26293$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.64 + 2.84i)3-s + (−10.0 − 17.4i)5-s + (15.8 + 9.49i)7-s + (8.10 + 14.0i)9-s + (−5.5 + 9.52i)11-s + 18.0·13-s + 66.2·15-s + (−3.37 + 5.83i)17-s + (−79.1 − 137. i)19-s + (−53.1 + 29.6i)21-s + (−42.5 − 73.6i)23-s + (−140. + 244. i)25-s − 141.·27-s − 49.3·29-s + (74.7 − 129. i)31-s + ⋯
L(s)  = 1  + (−0.316 + 0.547i)3-s + (−0.901 − 1.56i)5-s + (0.858 + 0.512i)7-s + (0.300 + 0.519i)9-s + (−0.150 + 0.261i)11-s + 0.384·13-s + 1.14·15-s + (−0.0480 + 0.0832i)17-s + (−0.955 − 1.65i)19-s + (−0.552 + 0.307i)21-s + (−0.385 − 0.667i)23-s + (−1.12 + 1.95i)25-s − 1.01·27-s − 0.316·29-s + (0.433 − 0.750i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.566 + 0.824i$
Analytic conductor: \(18.1725\)
Root analytic conductor: \(4.26293\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :3/2),\ -0.566 + 0.824i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7182602694\)
\(L(\frac12)\) \(\approx\) \(0.7182602694\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-15.8 - 9.49i)T \)
11 \( 1 + (5.5 - 9.52i)T \)
good3 \( 1 + (1.64 - 2.84i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (10.0 + 17.4i)T + (-62.5 + 108. i)T^{2} \)
13 \( 1 - 18.0T + 2.19e3T^{2} \)
17 \( 1 + (3.37 - 5.83i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (79.1 + 137. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42.5 + 73.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 49.3T + 2.43e4T^{2} \)
31 \( 1 + (-74.7 + 129. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (142. + 246. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 223.T + 6.89e4T^{2} \)
43 \( 1 + 38.0T + 7.95e4T^{2} \)
47 \( 1 + (205. + 355. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-178. + 309. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-374. + 647. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-62.5 - 108. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (318. - 552. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 176.T + 3.57e5T^{2} \)
73 \( 1 + (526. - 912. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (209. + 363. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.47e3T + 5.71e5T^{2} \)
89 \( 1 + (671. + 1.16e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.28e3T + 9.12e5T^{2} \)
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

−11.21109792475573018754807819648, −10.02077000342174524706854054578, −8.711797643467720210598921400168, −8.439345255179660065494956090141, −7.24947784241962891782023151590, −5.50666432525965895334548466707, −4.73335592894751859716707161286, −4.14726877480531587492696732148, −1.95048220957497969563436854669, −0.27963632265759171033911880686, 1.55792732092805029998163457571, 3.31192764581667646214187448450, 4.19300246788901096284359532530, 5.97668132775788877591964023319, 6.82718328442729286872384317509, 7.62466269490293310229623289331, 8.361930241918033591813977924928, 10.13693187619668251399261003479, 10.72417372729609724655431748761, 11.63483574366011021863439710382

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