↑

Properties

Label 2-69-23.22-c6-0-5
Degree $2$
Conductor $69$
Sign $-0.992 - 0.126i$
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  − 8.53·2-s − 15.5·3-s + 8.90·4-s + 233. i·5-s + 133.·6-s + 155. i·7-s + 470.·8-s + 243·9-s − 1.99e3i·10-s + 880. i·11-s − 138.·12-s + 3.10e3·13-s − 1.33e3i·14-s − 3.63e3i·15-s − 4.58e3·16-s + 5.58e3i·17-s + ⋯
L(s)  = 1  − 1.06·2-s − 0.577·3-s + 0.139·4-s + 1.86i·5-s + 0.616·6-s + 0.454i·7-s + 0.918·8-s + 0.333·9-s − 1.99i·10-s + 0.661i·11-s − 0.0803·12-s + 1.41·13-s − 0.485i·14-s − 1.07i·15-s − 1.11·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.992 - 0.126i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.992 - 0.126i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0420103 + 0.663433i\)
\(L(\frac12)\) \(\approx\) \(0.0420103 + 0.663433i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
23 \( 1 + (-1.20e4 - 1.53e3i)T \)
good2 \( 1 + 8.53T + 64T^{2} \)
5 \( 1 - 233. iT - 1.56e4T^{2} \)
7 \( 1 - 155. iT - 1.17e5T^{2} \)
11 \( 1 - 880. iT - 1.77e6T^{2} \)
13 \( 1 - 3.10e3T + 4.82e6T^{2} \)
17 \( 1 - 5.58e3iT - 2.41e7T^{2} \)
19 \( 1 - 9.60e3iT - 4.70e7T^{2} \)
29 \( 1 - 1.49e4T + 5.94e8T^{2} \)
31 \( 1 + 4.97e4T + 8.87e8T^{2} \)
37 \( 1 + 7.10e3iT - 2.56e9T^{2} \)
41 \( 1 - 3.96e4T + 4.75e9T^{2} \)
43 \( 1 - 9.71e4iT - 6.32e9T^{2} \)
47 \( 1 + 1.44e4T + 1.07e10T^{2} \)
53 \( 1 + 1.96e5iT - 2.21e10T^{2} \)
59 \( 1 - 1.77e5T + 4.21e10T^{2} \)
61 \( 1 + 3.43e5iT - 5.15e10T^{2} \)
67 \( 1 + 3.00e5iT - 9.04e10T^{2} \)
71 \( 1 + 1.04e5T + 1.28e11T^{2} \)
73 \( 1 + 4.66e5T + 1.51e11T^{2} \)
79 \( 1 - 7.80e5iT - 2.43e11T^{2} \)
83 \( 1 - 6.08e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.46e5iT - 4.96e11T^{2} \)
97 \( 1 + 6.91e5iT - 8.32e11T^{2} \)
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

−14.26027056608045844759834249514, −12.84317099260023322437387623871, −11.20622609788879728185554614866, −10.66125560642704636925651772052, −9.723774990550957913106117767206, −8.219884610030892995007107899632, −7.03550003045249966562048800822, −5.95249748188187839418463862615, −3.72192671026015171921448608823, −1.72741598789934248707010578017, 0.52330078859730317461896747364, 1.10713922981021821249392826871, 4.33301667507817415934820347294, 5.43448349140796824217536836504, 7.31855800370626997503750591521, 8.818465452890862909818031662840, 9.046888811291336122385185831554, 10.63751435422250912124531452834, 11.62804824262646054254506634853, 13.15656693418730258836859705879

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