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Properties

Label 2-29-29.12-c8-0-2
Degree $2$
Conductor $29$
Sign $0.997 + 0.0725i$
Analytic cond. $11.8139$
Root an. cond. $3.43714$
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  + (−15.2 − 15.2i)2-s + (93.1 + 93.1i)3-s + 211. i·4-s − 424. i·5-s − 2.84e3i·6-s + 996.·7-s + (−683. + 683. i)8-s + 1.07e4i·9-s + (−6.49e3 + 6.49e3i)10-s + (1.21e4 + 1.21e4i)11-s + (−1.96e4 + 1.96e4i)12-s − 1.25e4i·13-s + (−1.52e4 − 1.52e4i)14-s + (3.95e4 − 3.95e4i)15-s + 7.49e4·16-s + (−902. − 902. i)17-s + ⋯
L(s)  = 1  + (−0.955 − 0.955i)2-s + (1.14 + 1.14i)3-s + 0.825i·4-s − 0.679i·5-s − 2.19i·6-s + 0.415·7-s + (−0.166 + 0.166i)8-s + 1.64i·9-s + (−0.649 + 0.649i)10-s + (0.831 + 0.831i)11-s + (−0.949 + 0.949i)12-s − 0.437i·13-s + (−0.396 − 0.396i)14-s + (0.781 − 0.781i)15-s + 1.14·16-s + (−0.0108 − 0.0108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.997 + 0.0725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.997 + 0.0725i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.997 + 0.0725i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.67422 - 0.0608280i\)
\(L(\frac12)\) \(\approx\) \(1.67422 - 0.0608280i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (6.82e5 + 1.83e5i)T \)
good2 \( 1 + (15.2 + 15.2i)T + 256iT^{2} \)
3 \( 1 + (-93.1 - 93.1i)T + 6.56e3iT^{2} \)
5 \( 1 + 424. iT - 3.90e5T^{2} \)
7 \( 1 - 996.T + 5.76e6T^{2} \)
11 \( 1 + (-1.21e4 - 1.21e4i)T + 2.14e8iT^{2} \)
13 \( 1 + 1.25e4iT - 8.15e8T^{2} \)
17 \( 1 + (902. + 902. i)T + 6.97e9iT^{2} \)
19 \( 1 + (-8.44e4 - 8.44e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 3.63e5T + 7.83e10T^{2} \)
31 \( 1 + (-1.12e6 - 1.12e6i)T + 8.52e11iT^{2} \)
37 \( 1 + (-1.28e6 + 1.28e6i)T - 3.51e12iT^{2} \)
41 \( 1 + (4.33e5 - 4.33e5i)T - 7.98e12iT^{2} \)
43 \( 1 + (-6.49e5 - 6.49e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (3.20e6 - 3.20e6i)T - 2.38e13iT^{2} \)
53 \( 1 + 9.37e6T + 6.22e13T^{2} \)
59 \( 1 + 3.37e6T + 1.46e14T^{2} \)
61 \( 1 + (2.81e6 + 2.81e6i)T + 1.91e14iT^{2} \)
67 \( 1 + 3.39e7iT - 4.06e14T^{2} \)
71 \( 1 + 3.53e7iT - 6.45e14T^{2} \)
73 \( 1 + (3.20e7 - 3.20e7i)T - 8.06e14iT^{2} \)
79 \( 1 + (-3.31e7 - 3.31e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 1.04e7T + 2.25e15T^{2} \)
89 \( 1 + (6.19e7 + 6.19e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (1.03e8 - 1.03e8i)T - 7.83e15iT^{2} \)
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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

−15.19028471724201944743204829048, −14.31410020242010233877796265203, −12.50088214585660077266031663269, −11.02026075907439170341264102997, −9.762677848042427335340757392650, −9.120812170316527218134192998394, −8.054581080385798488754554075822, −4.76985968133799966000060046938, −3.13250026757599702043243446900, −1.43247997598572320726318585749, 1.08957077278164842998117565545, 3.05892711884766729087079966592, 6.47451775663189855070622122430, 7.32435277656372425434618488600, 8.429323734688964692326766289814, 9.353326828460528767893953257106, 11.47401272883754309793604482549, 13.21215849857301403737365197879, 14.36562015913511698438057526679, 15.11735090154992371472298340925

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