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Properties

Label 2-69-23.22-c6-0-13
Degree $2$
Conductor $69$
Sign $0.493 - 0.869i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.3·2-s − 15.5·3-s + 114.·4-s + 40.7i·5-s − 208.·6-s + 469. i·7-s + 675.·8-s + 243·9-s + 544. i·10-s + 1.56e3i·11-s − 1.78e3·12-s + 3.79e3·13-s + 6.27e3i·14-s − 635. i·15-s + 1.69e3·16-s − 9.02e3i·17-s + ⋯
L(s)  = 1  + 1.67·2-s − 0.577·3-s + 1.78·4-s + 0.326i·5-s − 0.964·6-s + 1.36i·7-s + 1.31·8-s + 0.333·9-s + 0.544i·10-s + 1.17i·11-s − 1.03·12-s + 1.72·13-s + 2.28i·14-s − 0.188i·15-s + 0.413·16-s − 1.83i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.493 - 0.869i$
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: \((2,\ 69,\ (\ :3),\ 0.493 - 0.869i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.44256 + 2.00551i\)
\(L(\frac12)\) \(\approx\) \(3.44256 + 2.00551i\)
\(L(4)\) 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 \( 1 + (6.00e3 - 1.05e4i)T \)
good2 \( 1 - 13.3T + 64T^{2} \)
5 \( 1 - 40.7iT - 1.56e4T^{2} \)
7 \( 1 - 469. iT - 1.17e5T^{2} \)
11 \( 1 - 1.56e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.79e3T + 4.82e6T^{2} \)
17 \( 1 + 9.02e3iT - 2.41e7T^{2} \)
19 \( 1 - 7.10e3iT - 4.70e7T^{2} \)
29 \( 1 + 3.65e3T + 5.94e8T^{2} \)
31 \( 1 + 2.75e4T + 8.87e8T^{2} \)
37 \( 1 + 7.51e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.16e3T + 4.75e9T^{2} \)
43 \( 1 + 7.70e4iT - 6.32e9T^{2} \)
47 \( 1 - 1.28e5T + 1.07e10T^{2} \)
53 \( 1 + 1.49e5iT - 2.21e10T^{2} \)
59 \( 1 + 5.59e4T + 4.21e10T^{2} \)
61 \( 1 - 1.67e5iT - 5.15e10T^{2} \)
67 \( 1 + 1.99e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.20e5T + 1.28e11T^{2} \)
73 \( 1 - 6.59e5T + 1.51e11T^{2} \)
79 \( 1 + 4.58e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.93e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.30e4iT - 4.96e11T^{2} \)
97 \( 1 - 8.56e5iT - 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.65506581632664842412341738261, −12.50622632437103251920455526416, −11.89015760192043964355339228446, −10.92720895233970921909925710536, −9.213126536517667681675381987691, −7.18618487164893973366640314838, −5.95293428725246473208849145275, −5.18946320133748025175708179255, −3.66714432920225969930290966284, −2.12557648180179629227571092154, 1.07131371376534420386286105074, 3.52517052029187335447792193180, 4.39155120708257403181402409360, 5.88324500626046990538626125985, 6.66061642999113385954422082868, 8.436767992226026046500850750239, 10.81190249253243351007543711143, 11.04040364675975881763374935890, 12.62643835188365202575382421656, 13.35152465184363432021123549269

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