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Properties

Label 2-29-29.17-c6-0-13
Degree $2$
Conductor $29$
Sign $-0.596 + 0.802i$
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  + (10.8 − 10.8i)2-s + (5.95 − 5.95i)3-s − 173. i·4-s + 50.8i·5-s − 129. i·6-s + 66.1·7-s + (−1.18e3 − 1.18e3i)8-s + 658. i·9-s + (554. + 554. i)10-s + (1.33e3 − 1.33e3i)11-s + (−1.03e3 − 1.03e3i)12-s + 2.39e3i·13-s + (720. − 720. i)14-s + (302. + 302. i)15-s − 1.48e4·16-s + (−2.03e3 + 2.03e3i)17-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s + (0.220 − 0.220i)3-s − 2.70i·4-s + 0.407i·5-s − 0.600i·6-s + 0.192·7-s + (−2.32 − 2.32i)8-s + 0.902i·9-s + (0.554 + 0.554i)10-s + (1.00 − 1.00i)11-s + (−0.596 − 0.596i)12-s + 1.09i·13-s + (0.262 − 0.262i)14-s + (0.0897 + 0.0897i)15-s − 3.61·16-s + (−0.414 + 0.414i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.596 + 0.802i$
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: \((2,\ 29,\ (\ :3),\ -0.596 + 0.802i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.50449 - 2.99171i\)
\(L(\frac12)\) \(\approx\) \(1.50449 - 2.99171i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (1.05e4 - 2.19e4i)T \)
good2 \( 1 + (-10.8 + 10.8i)T - 64iT^{2} \)
3 \( 1 + (-5.95 + 5.95i)T - 729iT^{2} \)
5 \( 1 - 50.8iT - 1.56e4T^{2} \)
7 \( 1 - 66.1T + 1.17e5T^{2} \)
11 \( 1 + (-1.33e3 + 1.33e3i)T - 1.77e6iT^{2} \)
13 \( 1 - 2.39e3iT - 4.82e6T^{2} \)
17 \( 1 + (2.03e3 - 2.03e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (-5.91e3 + 5.91e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 558.T + 1.48e8T^{2} \)
31 \( 1 + (4.77e3 - 4.77e3i)T - 8.87e8iT^{2} \)
37 \( 1 + (4.01e4 + 4.01e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-9.05e4 - 9.05e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (6.28e4 - 6.28e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (1.00e5 + 1.00e5i)T + 1.07e10iT^{2} \)
53 \( 1 + 1.97e5T + 2.21e10T^{2} \)
59 \( 1 + 1.78e5T + 4.21e10T^{2} \)
61 \( 1 + (8.31e4 - 8.31e4i)T - 5.15e10iT^{2} \)
67 \( 1 + 2.14e5iT - 9.04e10T^{2} \)
71 \( 1 - 1.16e5iT - 1.28e11T^{2} \)
73 \( 1 + (-3.61e5 - 3.61e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (-2.17e5 + 2.17e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 3.65e5T + 3.26e11T^{2} \)
89 \( 1 + (-7.37e5 + 7.37e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-5.62e5 - 5.62e5i)T + 8.32e11iT^{2} \)
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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

−14.57391011010869272209681962447, −13.99897822616137086936306136481, −12.97372062847253690129282907881, −11.47878297207849178987611987072, −10.91950127826657835877445423664, −9.198365468698997438514980883595, −6.54222218334736115804242506253, −4.82479086794664653827226346697, −3.21743026361259931645582147318, −1.59739101953323923413798230890, 3.53856764971934501624361027685, 4.90271030948439929060527937347, 6.40103439262351791316921521699, 7.76611418993651229885587057706, 9.246267675691989840012383198568, 11.95133108295555382834396861970, 12.74289474025802251203147484535, 14.14094352294208822205932316404, 14.98653036901431047410815635341, 15.80850372498803048475737754909

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