Properties

Label 2-69-23.13-c5-0-19
Degree $2$
Conductor $69$
Sign $-0.820 + 0.571i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.220 − 0.254i)2-s + (7.57 + 4.86i)3-s + (4.53 − 31.5i)4-s + (21.3 − 46.7i)5-s + (−0.430 − 2.99i)6-s + (−219. − 64.3i)7-s + (−18.0 + 11.6i)8-s + (33.6 + 73.6i)9-s + (−16.5 + 4.87i)10-s + (−314. + 363. i)11-s + (187. − 216. i)12-s + (−42.3 + 12.4i)13-s + (31.9 + 69.9i)14-s + (389. − 250. i)15-s + (−972. − 285. i)16-s + (−95.5 − 664. i)17-s + ⋯
L(s)  = 1  + (−0.0389 − 0.0449i)2-s + (0.485 + 0.312i)3-s + (0.141 − 0.986i)4-s + (0.382 − 0.836i)5-s + (−0.00488 − 0.0339i)6-s + (−1.69 − 0.496i)7-s + (−0.0998 + 0.0641i)8-s + (0.138 + 0.303i)9-s + (−0.0524 + 0.0154i)10-s + (−0.783 + 0.904i)11-s + (0.376 − 0.434i)12-s + (−0.0694 + 0.0203i)13-s + (0.0435 + 0.0953i)14-s + (0.446 − 0.287i)15-s + (−0.949 − 0.278i)16-s + (−0.0801 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.820 + 0.571i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ -0.820 + 0.571i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.336721 - 1.07158i\)
\(L(\frac12)\) \(\approx\) \(0.336721 - 1.07158i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.57 - 4.86i)T \)
23 \( 1 + (65.8 + 2.53e3i)T \)
good2 \( 1 + (0.220 + 0.254i)T + (-4.55 + 31.6i)T^{2} \)
5 \( 1 + (-21.3 + 46.7i)T + (-2.04e3 - 2.36e3i)T^{2} \)
7 \( 1 + (219. + 64.3i)T + (1.41e4 + 9.08e3i)T^{2} \)
11 \( 1 + (314. - 363. i)T + (-2.29e4 - 1.59e5i)T^{2} \)
13 \( 1 + (42.3 - 12.4i)T + (3.12e5 - 2.00e5i)T^{2} \)
17 \( 1 + (95.5 + 664. i)T + (-1.36e6 + 4.00e5i)T^{2} \)
19 \( 1 + (6.76 - 47.0i)T + (-2.37e6 - 6.97e5i)T^{2} \)
29 \( 1 + (660. + 4.59e3i)T + (-1.96e7 + 5.77e6i)T^{2} \)
31 \( 1 + (-2.96e3 + 1.90e3i)T + (1.18e7 - 2.60e7i)T^{2} \)
37 \( 1 + (5.15e3 + 1.12e4i)T + (-4.54e7 + 5.24e7i)T^{2} \)
41 \( 1 + (-5.91e3 + 1.29e4i)T + (-7.58e7 - 8.75e7i)T^{2} \)
43 \( 1 + (-1.84e4 - 1.18e4i)T + (6.10e7 + 1.33e8i)T^{2} \)
47 \( 1 - 7.82e3T + 2.29e8T^{2} \)
53 \( 1 + (2.61e4 + 7.66e3i)T + (3.51e8 + 2.26e8i)T^{2} \)
59 \( 1 + (-9.10e3 + 2.67e3i)T + (6.01e8 - 3.86e8i)T^{2} \)
61 \( 1 + (2.16e4 - 1.38e4i)T + (3.50e8 - 7.68e8i)T^{2} \)
67 \( 1 + (-3.08e4 - 3.56e4i)T + (-1.92e8 + 1.33e9i)T^{2} \)
71 \( 1 + (7.49e3 + 8.65e3i)T + (-2.56e8 + 1.78e9i)T^{2} \)
73 \( 1 + (-3.95e3 + 2.74e4i)T + (-1.98e9 - 5.84e8i)T^{2} \)
79 \( 1 + (6.55e4 - 1.92e4i)T + (2.58e9 - 1.66e9i)T^{2} \)
83 \( 1 + (4.82e4 + 1.05e5i)T + (-2.57e9 + 2.97e9i)T^{2} \)
89 \( 1 + (-6.63e4 - 4.26e4i)T + (2.31e9 + 5.07e9i)T^{2} \)
97 \( 1 + (-2.43e4 + 5.33e4i)T + (-5.62e9 - 6.48e9i)T^{2} \)
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

−13.29406736859257842195982350853, −12.54549946558578331063356203684, −10.60428892491536397057011204073, −9.768583046170640922830573534089, −9.142596565664419078346700510677, −7.22926288762815539610582270851, −5.86146548679282015405115156281, −4.46136795647076888830097435334, −2.46893270917688249475328170566, −0.44923621503531846550605862515, 2.73395768085253973047506837896, 3.35262149785667960441763963529, 6.08474490942004361181395424471, 7.03148489361140014679139514204, 8.359745213164002073364729195374, 9.509330034488059934602249844816, 10.78712915223891276706114947726, 12.30255199779173290728691995679, 13.09069619140327677145211563154, 13.91627684340420424005939728024

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