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Properties

Label 2-29-29.17-c6-0-10
Degree $2$
Conductor $29$
Sign $0.187 + 0.982i$
Analytic cond. $6.67156$
Root an. cond. $2.58293$
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  + (−5.36 + 5.36i)2-s + (27.0 − 27.0i)3-s + 6.49i·4-s − 130. i·5-s + 289. i·6-s − 330.·7-s + (−378. − 378. i)8-s − 730. i·9-s + (701. + 701. i)10-s + (869. − 869. i)11-s + (175. + 175. i)12-s − 2.44e3i·13-s + (1.77e3 − 1.77e3i)14-s + (−3.53e3 − 3.53e3i)15-s + 3.63e3·16-s + (3.36e3 − 3.36e3i)17-s + ⋯
L(s)  = 1  + (−0.670 + 0.670i)2-s + (1.00 − 1.00i)3-s + 0.101i·4-s − 1.04i·5-s + 1.34i·6-s − 0.964·7-s + (−0.738 − 0.738i)8-s − 1.00i·9-s + (0.701 + 0.701i)10-s + (0.653 − 0.653i)11-s + (0.101 + 0.101i)12-s − 1.11i·13-s + (0.646 − 0.646i)14-s + (−1.04 − 1.04i)15-s + 0.888·16-s + (0.684 − 0.684i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(6.67156\)
Root analytic conductor: \(2.58293\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :3),\ 0.187 + 0.982i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.936433 - 0.774405i\)
\(L(\frac12)\) \(\approx\) \(0.936433 - 0.774405i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-9.02e3 - 2.26e4i)T \)
good2 \( 1 + (5.36 - 5.36i)T - 64iT^{2} \)
3 \( 1 + (-27.0 + 27.0i)T - 729iT^{2} \)
5 \( 1 + 130. iT - 1.56e4T^{2} \)
7 \( 1 + 330.T + 1.17e5T^{2} \)
11 \( 1 + (-869. + 869. i)T - 1.77e6iT^{2} \)
13 \( 1 + 2.44e3iT - 4.82e6T^{2} \)
17 \( 1 + (-3.36e3 + 3.36e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (3.39e3 - 3.39e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.41e4T + 1.48e8T^{2} \)
31 \( 1 + (-1.89e4 + 1.89e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (3.54e4 + 3.54e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-7.50e4 - 7.50e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (3.89e4 - 3.89e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-1.05e5 - 1.05e5i)T + 1.07e10iT^{2} \)
53 \( 1 + 1.93e5T + 2.21e10T^{2} \)
59 \( 1 - 3.14e5T + 4.21e10T^{2} \)
61 \( 1 + (-1.32e5 + 1.32e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 3.54e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.07e5iT - 1.28e11T^{2} \)
73 \( 1 + (-1.22e5 - 1.22e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (1.67e5 - 1.67e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 2.87e5T + 3.26e11T^{2} \)
89 \( 1 + (-2.70e5 + 2.70e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (6.78e5 + 6.78e5i)T + 8.32e11iT^{2} \)
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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

−15.91613906399632341454922462388, −14.25040141972330300095289954697, −12.92035158433619616750241540144, −12.38924713601462895697998663378, −9.598887181682376427604680089888, −8.546604721380688849894150025066, −7.73786427299983452778662794599, −6.24356342405049405557067470667, −3.23850043070273343914037006254, −0.75526565698052628830274414434, 2.39200262467867643518233602875, 3.82520778119604997996489318089, 6.55679992031950855477964286208, 8.730813486281683133297668571923, 9.813028278176951368563045361278, 10.35648806133190380621666919757, 11.92696906363558205130830929780, 14.08233882903207517486002379622, 14.78798615391598857293016859130, 15.83164511670007057642576171606

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