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Properties

Label 2-69-23.22-c8-0-5
Degree $2$
Conductor $69$
Sign $-0.737 + 0.675i$
Analytic cond. $28.1091$
Root an. cond. $5.30180$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.59·2-s − 46.7·3-s − 249.·4-s + 1.02e3i·5-s − 121.·6-s + 2.63e3i·7-s − 1.31e3·8-s + 2.18e3·9-s + 2.64e3i·10-s + 1.90e4i·11-s + 1.16e4·12-s − 3.55e3·13-s + 6.82e3i·14-s − 4.77e4i·15-s + 6.04e4·16-s + 3.50e4i·17-s + ⋯
L(s)  = 1  + 0.162·2-s − 0.577·3-s − 0.973·4-s + 1.63i·5-s − 0.0935·6-s + 1.09i·7-s − 0.319·8-s + 0.333·9-s + 0.264i·10-s + 1.30i·11-s + 0.562·12-s − 0.124·13-s + 0.177i·14-s − 0.943i·15-s + 0.921·16-s + 0.420i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.737 + 0.675i$
Analytic conductor: \(28.1091\)
Root analytic conductor: \(5.30180\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :4),\ -0.737 + 0.675i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.274342 - 0.705645i\)
\(L(\frac12)\) \(\approx\) \(0.274342 - 0.705645i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 46.7T \)
23 \( 1 + (2.06e5 - 1.89e5i)T \)
good2 \( 1 - 2.59T + 256T^{2} \)
5 \( 1 - 1.02e3iT - 3.90e5T^{2} \)
7 \( 1 - 2.63e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.90e4iT - 2.14e8T^{2} \)
13 \( 1 + 3.55e3T + 8.15e8T^{2} \)
17 \( 1 - 3.50e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.17e4iT - 1.69e10T^{2} \)
29 \( 1 + 1.36e6T + 5.00e11T^{2} \)
31 \( 1 - 1.61e6T + 8.52e11T^{2} \)
37 \( 1 - 1.02e6iT - 3.51e12T^{2} \)
41 \( 1 - 1.05e6T + 7.98e12T^{2} \)
43 \( 1 + 3.42e6iT - 1.16e13T^{2} \)
47 \( 1 + 5.18e5T + 2.38e13T^{2} \)
53 \( 1 + 5.78e6iT - 6.22e13T^{2} \)
59 \( 1 - 6.98e6T + 1.46e14T^{2} \)
61 \( 1 - 1.17e7iT - 1.91e14T^{2} \)
67 \( 1 + 2.98e7iT - 4.06e14T^{2} \)
71 \( 1 - 2.70e7T + 6.45e14T^{2} \)
73 \( 1 - 3.76e7T + 8.06e14T^{2} \)
79 \( 1 + 2.07e7iT - 1.51e15T^{2} \)
83 \( 1 - 6.66e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.93e7iT - 3.93e15T^{2} \)
97 \( 1 - 8.17e7iT - 7.83e15T^{2} \)
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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

−13.81544579177541250073936831678, −12.52754759772889275562378285011, −11.63579330420167916016608770757, −10.28571967741999936713627829333, −9.482662908401396729371594436301, −7.79843485868614971082555921472, −6.45831709525317796973244998672, −5.33129188572451932467014326249, −3.79147956867383933794455186146, −2.19830782260627503800666254928, 0.33851891304527917224783261995, 0.943071454667571142136353332189, 3.92099504798054342464365848336, 4.81844434507599828094330141386, 5.90404537183628489827167980037, 7.88516464605815722837965793134, 8.898892211804917382997786493983, 9.973079170671723184626408707150, 11.39414358480121591632474289251, 12.63539310516646744047165637009

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