↑

Properties

Label 2-69-23.22-c4-0-15
Degree $2$
Conductor $69$
Sign $0.785 + 0.618i$
Analytic cond. $7.13252$
Root an. cond. $2.67067$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 7.44·2-s − 5.19·3-s + 39.3·4-s − 49.1i·5-s − 38.6·6-s + 11.1i·7-s + 173.·8-s + 27·9-s − 365. i·10-s + 158. i·11-s − 204.·12-s + 3.48·13-s + 83.1i·14-s + 255. i·15-s + 664.·16-s + 273. i·17-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.46·4-s − 1.96i·5-s − 1.07·6-s + 0.228i·7-s + 2.71·8-s + 0.333·9-s − 3.65i·10-s + 1.30i·11-s − 1.42·12-s + 0.0206·13-s + 0.424i·14-s + 1.13i·15-s + 2.59·16-s + 0.946i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.785 + 0.618i$
Analytic conductor: \(7.13252\)
Root analytic conductor: \(2.67067\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :2),\ 0.785 + 0.618i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.72011 - 1.28814i\)
\(L(\frac12)\) \(\approx\) \(3.72011 - 1.28814i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
23 \( 1 + (-415. - 327. i)T \)
good2 \( 1 - 7.44T + 16T^{2} \)
5 \( 1 + 49.1iT - 625T^{2} \)
7 \( 1 - 11.1iT - 2.40e3T^{2} \)
11 \( 1 - 158. iT - 1.46e4T^{2} \)
13 \( 1 - 3.48T + 2.85e4T^{2} \)
17 \( 1 - 273. iT - 8.35e4T^{2} \)
19 \( 1 + 195. iT - 1.30e5T^{2} \)
29 \( 1 + 1.09e3T + 7.07e5T^{2} \)
31 \( 1 - 362.T + 9.23e5T^{2} \)
37 \( 1 + 1.38e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.11e3T + 2.82e6T^{2} \)
43 \( 1 - 2.50e3iT - 3.41e6T^{2} \)
47 \( 1 + 420.T + 4.87e6T^{2} \)
53 \( 1 - 2.75e3iT - 7.89e6T^{2} \)
59 \( 1 + 3.19e3T + 1.21e7T^{2} \)
61 \( 1 + 3.44e3iT - 1.38e7T^{2} \)
67 \( 1 + 5.96e3iT - 2.01e7T^{2} \)
71 \( 1 - 3.42e3T + 2.54e7T^{2} \)
73 \( 1 + 4.87e3T + 2.83e7T^{2} \)
79 \( 1 + 968. iT - 3.89e7T^{2} \)
83 \( 1 + 6.56e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.07e3iT - 6.27e7T^{2} \)
97 \( 1 + 9.57e3iT - 8.85e7T^{2} \)
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

−13.44922244572009487042033686445, −12.66637481974488548900637556172, −12.31659686896700612923632602444, −11.14151747702566012196659302370, −9.375384896190211802493524581614, −7.54261182047013717314606973211, −5.91735340789742119555653815882, −4.97675232260453646821136396756, −4.19318012631561663547027071249, −1.69400351884170572059521037629, 2.72203134162555392545633792363, 3.76441642286832140854268859707, 5.57274772480198352262860967788, 6.50969599387931196209903886523, 7.36286202268686025941805832848, 10.39674281987047989654400598126, 11.15496437923083597070155916851, 11.78482618964338469338886058660, 13.33713863136621650983799874253, 14.07156786783645470601573825900

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