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Properties

Label 2-69-23.22-c8-0-29
Degree $2$
Conductor $69$
Sign $0.483 + 0.875i$
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  + 29.5·2-s − 46.7·3-s + 619.·4-s − 109. i·5-s − 1.38e3·6-s − 3.84e3i·7-s + 1.07e4·8-s + 2.18e3·9-s − 3.25e3i·10-s + 6.68e3i·11-s − 2.89e4·12-s − 4.93e3·13-s − 1.13e5i·14-s + 5.13e3i·15-s + 1.59e5·16-s − 1.56e5i·17-s + ⋯
L(s)  = 1  + 1.84·2-s − 0.577·3-s + 2.41·4-s − 0.175i·5-s − 1.06·6-s − 1.60i·7-s + 2.62·8-s + 0.333·9-s − 0.325i·10-s + 0.456i·11-s − 1.39·12-s − 0.172·13-s − 2.96i·14-s + 0.101i·15-s + 2.43·16-s − 1.87i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.483 + 0.875i$
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: \((2,\ 69,\ (\ :4),\ 0.483 + 0.875i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(4.45914 - 2.63155i\)
\(L(\frac12)\) \(\approx\) \(4.45914 - 2.63155i\)
\(L(5)\) 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 \( 1 + (-1.35e5 - 2.44e5i)T \)
good2 \( 1 - 29.5T + 256T^{2} \)
5 \( 1 + 109. iT - 3.90e5T^{2} \)
7 \( 1 + 3.84e3iT - 5.76e6T^{2} \)
11 \( 1 - 6.68e3iT - 2.14e8T^{2} \)
13 \( 1 + 4.93e3T + 8.15e8T^{2} \)
17 \( 1 + 1.56e5iT - 6.97e9T^{2} \)
19 \( 1 + 9.59e4iT - 1.69e10T^{2} \)
29 \( 1 + 1.29e5T + 5.00e11T^{2} \)
31 \( 1 - 1.37e6T + 8.52e11T^{2} \)
37 \( 1 - 1.04e6iT - 3.51e12T^{2} \)
41 \( 1 + 1.11e6T + 7.98e12T^{2} \)
43 \( 1 - 1.13e6iT - 1.16e13T^{2} \)
47 \( 1 + 7.33e6T + 2.38e13T^{2} \)
53 \( 1 - 1.73e6iT - 6.22e13T^{2} \)
59 \( 1 - 1.08e7T + 1.46e14T^{2} \)
61 \( 1 - 1.87e7iT - 1.91e14T^{2} \)
67 \( 1 - 2.21e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.81e7T + 6.45e14T^{2} \)
73 \( 1 - 1.60e7T + 8.06e14T^{2} \)
79 \( 1 - 1.09e7iT - 1.51e15T^{2} \)
83 \( 1 + 2.97e7iT - 2.25e15T^{2} \)
89 \( 1 - 1.19e8iT - 3.93e15T^{2} \)
97 \( 1 - 9.05e7iT - 7.83e15T^{2} \)
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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

−13.21567999939484619019100119403, −11.88851251753228746312842825222, −11.16179124798098400625752403567, −9.977399148899037733577462783153, −7.31548339280779844685055795419, −6.75503232296422647144351784681, −5.06865789079737570333906964137, −4.44126711667876473687621483312, −2.98733942620547095523933632873, −1.00450565571738657679078467878, 1.96072551494127252190030949200, 3.27080925418207313697065348348, 4.77484736916112348981570237418, 5.85976263991676042065759011305, 6.48147430625542596487136720781, 8.384794551428031585503166373498, 10.45668887348698283387168662968, 11.52009725548303338356417258900, 12.40070697493169322154762976684, 12.94916860664852132386335201280

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