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Properties

Label 2-29-29.12-c6-0-13
Degree $2$
Conductor $29$
Sign $-0.526 - 0.850i$
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  + (−1.83 − 1.83i)2-s + (−32.7 − 32.7i)3-s − 57.2i·4-s − 4.82i·5-s + 120. i·6-s + 52.1·7-s + (−222. + 222. i)8-s + 1.42e3i·9-s + (−8.86 + 8.86i)10-s + (−196. − 196. i)11-s + (−1.87e3 + 1.87e3i)12-s − 1.41e3i·13-s + (−95.8 − 95.8i)14-s + (−158. + 158. i)15-s − 2.84e3·16-s + (4.23e3 + 4.23e3i)17-s + ⋯
L(s)  = 1  + (−0.229 − 0.229i)2-s + (−1.21 − 1.21i)3-s − 0.894i·4-s − 0.0386i·5-s + 0.557i·6-s + 0.152·7-s + (−0.434 + 0.434i)8-s + 1.94i·9-s + (−0.00886 + 0.00886i)10-s + (−0.147 − 0.147i)11-s + (−1.08 + 1.08i)12-s − 0.645i·13-s + (−0.0349 − 0.0349i)14-s + (−0.0469 + 0.0469i)15-s − 0.694·16-s + (0.862 + 0.862i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.526 - 0.850i$
Analytic conductor: \(6.67156\)
Root analytic conductor: \(2.58293\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :3),\ -0.526 - 0.850i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.179640 + 0.322672i\)
\(L(\frac12)\) \(\approx\) \(0.179640 + 0.322672i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (8.69e3 + 2.27e4i)T \)
good2 \( 1 + (1.83 + 1.83i)T + 64iT^{2} \)
3 \( 1 + (32.7 + 32.7i)T + 729iT^{2} \)
5 \( 1 + 4.82iT - 1.56e4T^{2} \)
7 \( 1 - 52.1T + 1.17e5T^{2} \)
11 \( 1 + (196. + 196. i)T + 1.77e6iT^{2} \)
13 \( 1 + 1.41e3iT - 4.82e6T^{2} \)
17 \( 1 + (-4.23e3 - 4.23e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (810. + 810. i)T + 4.70e7iT^{2} \)
23 \( 1 + 1.74e4T + 1.48e8T^{2} \)
31 \( 1 + (3.30e4 + 3.30e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (5.53e4 - 5.53e4i)T - 2.56e9iT^{2} \)
41 \( 1 + (3.46e4 - 3.46e4i)T - 4.75e9iT^{2} \)
43 \( 1 + (5.33e3 + 5.33e3i)T + 6.32e9iT^{2} \)
47 \( 1 + (2.46e4 - 2.46e4i)T - 1.07e10iT^{2} \)
53 \( 1 - 9.68e4T + 2.21e10T^{2} \)
59 \( 1 - 2.87e5T + 4.21e10T^{2} \)
61 \( 1 + (9.71e4 + 9.71e4i)T + 5.15e10iT^{2} \)
67 \( 1 + 3.93e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.61e5iT - 1.28e11T^{2} \)
73 \( 1 + (4.53e4 - 4.53e4i)T - 1.51e11iT^{2} \)
79 \( 1 + (2.50e5 + 2.50e5i)T + 2.43e11iT^{2} \)
83 \( 1 - 7.62e4T + 3.26e11T^{2} \)
89 \( 1 + (-7.53e5 - 7.53e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (1.63e5 - 1.63e5i)T - 8.32e11iT^{2} \)
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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

−14.94951202184424109142684305815, −13.49659689437330627488711252790, −12.30136633528362105012612141878, −11.21141112677663595789750010645, −10.15214937278652375865936434363, −7.996995883914091917537302173734, −6.34529619901552465541717670988, −5.38387970902339807098232764433, −1.68484455254543224001095821821, −0.25533647975877853500794403772, 3.75830151500749487011124434142, 5.26151666290902168156912122242, 7.01398817156474018861653604794, 8.936354421723208784493885902660, 10.24813401893568812595665221331, 11.54517783023401596669573685293, 12.44255968924127002324803128281, 14.46992390938123777700239360342, 16.16378439439254031252782893767, 16.28130503157786788538979346206

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