↑

Properties

Label 2-69-3.2-c6-0-13
Degree $2$
Conductor $69$
Sign $-0.524 - 0.851i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.19i·2-s + (−22.9 + 14.1i)3-s + 62.5·4-s + 242. i·5-s + (16.9 + 27.4i)6-s + 379.·7-s − 151. i·8-s + (327. − 651. i)9-s + 289.·10-s + 415. i·11-s + (−1.43e3 + 886. i)12-s − 2.68e3·13-s − 453. i·14-s + (−3.44e3 − 5.58e3i)15-s + 3.82e3·16-s + 6.15e3i·17-s + ⋯
L(s)  = 1  − 0.149i·2-s + (−0.851 + 0.524i)3-s + 0.977·4-s + 1.94i·5-s + (0.0783 + 0.126i)6-s + 1.10·7-s − 0.295i·8-s + (0.448 − 0.893i)9-s + 0.289·10-s + 0.312i·11-s + (−0.832 + 0.513i)12-s − 1.22·13-s − 0.165i·14-s + (−1.02 − 1.65i)15-s + 0.933·16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.524 - 0.851i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.524 - 0.851i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.832400 + 1.49141i\)
\(L(\frac12)\) \(\approx\) \(0.832400 + 1.49141i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (22.9 - 14.1i)T \)
23 \( 1 + 2.53e3iT \)
good2 \( 1 + 1.19iT - 64T^{2} \)
5 \( 1 - 242. iT - 1.56e4T^{2} \)
7 \( 1 - 379.T + 1.17e5T^{2} \)
11 \( 1 - 415. iT - 1.77e6T^{2} \)
13 \( 1 + 2.68e3T + 4.82e6T^{2} \)
17 \( 1 - 6.15e3iT - 2.41e7T^{2} \)
19 \( 1 - 8.44e3T + 4.70e7T^{2} \)
29 \( 1 - 1.50e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.96e4T + 8.87e8T^{2} \)
37 \( 1 + 4.48e4T + 2.56e9T^{2} \)
41 \( 1 + 6.75e4iT - 4.75e9T^{2} \)
43 \( 1 + 2.37e4T + 6.32e9T^{2} \)
47 \( 1 + 8.41e3iT - 1.07e10T^{2} \)
53 \( 1 - 3.11e4iT - 2.21e10T^{2} \)
59 \( 1 - 7.48e4iT - 4.21e10T^{2} \)
61 \( 1 - 4.22e5T + 5.15e10T^{2} \)
67 \( 1 - 1.96e5T + 9.04e10T^{2} \)
71 \( 1 + 7.01e4iT - 1.28e11T^{2} \)
73 \( 1 - 7.08e4T + 1.51e11T^{2} \)
79 \( 1 - 2.02e5T + 2.43e11T^{2} \)
83 \( 1 - 1.38e4iT - 3.26e11T^{2} \)
89 \( 1 + 2.27e5iT - 4.96e11T^{2} \)
97 \( 1 - 3.82e5T + 8.32e11T^{2} \)
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

−14.49129971572631780430271774938, −12.23366860427687877682746345796, −11.39957885464254867162955484709, −10.68462970877691024700571985422, −10.00427393911902155773854274933, −7.51941796684527321135921563096, −6.79504284556592163196035036999, −5.46395466565024012563607553362, −3.53862536511770281549257424454, −1.99954442917745362827365157167, 0.73594621411867571588042772897, 1.87173890837970847573821338110, 4.93541105454454904191644420710, 5.44114155253337806071374051639, 7.29892522351243186608073792677, 8.123430159496859120830866670065, 9.710196096920618254600694899600, 11.50196029262853085486936539902, 11.82036579333374998216931736934, 12.84207808571095951329589480884

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