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Properties

Label 2-29-29.17-c6-0-5
Degree $2$
Conductor $29$
Sign $-0.0599 - 0.998i$
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  + (−6.90 + 6.90i)2-s + (11.2 − 11.2i)3-s − 31.3i·4-s − 10.9i·5-s + 155. i·6-s + 617.·7-s + (−225. − 225. i)8-s + 476. i·9-s + (75.5 + 75.5i)10-s + (−913. + 913. i)11-s + (−352. − 352. i)12-s + 3.60e3i·13-s + (−4.26e3 + 4.26e3i)14-s + (−122. − 122. i)15-s + 5.11e3·16-s + (3.54e3 − 3.54e3i)17-s + ⋯
L(s)  = 1  + (−0.863 + 0.863i)2-s + (0.416 − 0.416i)3-s − 0.489i·4-s − 0.0875i·5-s + 0.718i·6-s + 1.80·7-s + (−0.440 − 0.440i)8-s + 0.653i·9-s + (0.0755 + 0.0755i)10-s + (−0.686 + 0.686i)11-s + (−0.203 − 0.203i)12-s + 1.64i·13-s + (−1.55 + 1.55i)14-s + (−0.0364 − 0.0364i)15-s + 1.24·16-s + (0.721 − 0.721i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0599 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.0599 - 0.998i$
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: \((2,\ 29,\ (\ :3),\ -0.0599 - 0.998i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.868491 + 0.922198i\)
\(L(\frac12)\) \(\approx\) \(0.868491 + 0.922198i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (6.03e3 + 2.36e4i)T \)
good2 \( 1 + (6.90 - 6.90i)T - 64iT^{2} \)
3 \( 1 + (-11.2 + 11.2i)T - 729iT^{2} \)
5 \( 1 + 10.9iT - 1.56e4T^{2} \)
7 \( 1 - 617.T + 1.17e5T^{2} \)
11 \( 1 + (913. - 913. i)T - 1.77e6iT^{2} \)
13 \( 1 - 3.60e3iT - 4.82e6T^{2} \)
17 \( 1 + (-3.54e3 + 3.54e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (6.39e3 - 6.39e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 1.63e4T + 1.48e8T^{2} \)
31 \( 1 + (2.14e4 - 2.14e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (3.44e4 + 3.44e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (-8.95e3 - 8.95e3i)T + 4.75e9iT^{2} \)
43 \( 1 + (-3.87e4 + 3.87e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (2.47e4 + 2.47e4i)T + 1.07e10iT^{2} \)
53 \( 1 + 1.45e5T + 2.21e10T^{2} \)
59 \( 1 + 1.66e5T + 4.21e10T^{2} \)
61 \( 1 + (-6.04e4 + 6.04e4i)T - 5.15e10iT^{2} \)
67 \( 1 + 4.57e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.28e5iT - 1.28e11T^{2} \)
73 \( 1 + (-1.43e5 - 1.43e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (-1.71e5 + 1.71e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 2.29e5T + 3.26e11T^{2} \)
89 \( 1 + (-4.89e5 + 4.89e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (1.35e5 + 1.35e5i)T + 8.32e11iT^{2} \)
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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

−16.42876010980227621153117303800, −14.91186888331401727822466378286, −14.09687675460723722232850060965, −12.39152598049460940929578706900, −10.84740972502645450380124179551, −9.029253758053146415937541070008, −7.981823814367162717565279702447, −7.16238533578037428297547651895, −4.91046371178282054962470107743, −1.79006716738857399359866172854, 0.992596777936299669308027359414, 2.95350273292489078363317406687, 5.27758357175011539055075377160, 8.064264885885476622376025193859, 8.870299321544692908898831654817, 10.56982390711885506346822277912, 11.06842206981423031335370588384, 12.70037914998882962916291863377, 14.67607800217720027325041502941, 15.11118093139993273615291412134

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