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Properties

Label 2-29-29.12-c8-0-17
Degree $2$
Conductor $29$
Sign $-0.688 - 0.725i$
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  + (4.53 + 4.53i)2-s + (−83.7 − 83.7i)3-s − 214. i·4-s − 3.13i·5-s − 759. i·6-s − 1.56e3·7-s + (2.13e3 − 2.13e3i)8-s + 7.47e3i·9-s + (14.2 − 14.2i)10-s + (1.41e3 + 1.41e3i)11-s + (−1.80e4 + 1.80e4i)12-s + 5.35e4i·13-s + (−7.10e3 − 7.10e3i)14-s + (−262. + 262. i)15-s − 3.56e4·16-s + (−4.61e4 − 4.61e4i)17-s + ⋯
L(s)  = 1  + (0.283 + 0.283i)2-s + (−1.03 − 1.03i)3-s − 0.839i·4-s − 0.00501i·5-s − 0.586i·6-s − 0.653·7-s + (0.521 − 0.521i)8-s + 1.13i·9-s + (0.00142 − 0.00142i)10-s + (0.0966 + 0.0966i)11-s + (−0.868 + 0.868i)12-s + 1.87i·13-s + (−0.185 − 0.185i)14-s + (−0.00519 + 0.00519i)15-s − 0.544·16-s + (−0.552 − 0.552i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.688 - 0.725i$
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: \((2,\ 29,\ (\ :4),\ -0.688 - 0.725i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.0568446 + 0.132258i\)
\(L(\frac12)\) \(\approx\) \(0.0568446 + 0.132258i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.80e5 - 5.95e5i)T \)
good2 \( 1 + (-4.53 - 4.53i)T + 256iT^{2} \)
3 \( 1 + (83.7 + 83.7i)T + 6.56e3iT^{2} \)
5 \( 1 + 3.13iT - 3.90e5T^{2} \)
7 \( 1 + 1.56e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.41e3 - 1.41e3i)T + 2.14e8iT^{2} \)
13 \( 1 - 5.35e4iT - 8.15e8T^{2} \)
17 \( 1 + (4.61e4 + 4.61e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (5.23e4 + 5.23e4i)T + 1.69e10iT^{2} \)
23 \( 1 + 2.15e5T + 7.83e10T^{2} \)
31 \( 1 + (-9.66e4 - 9.66e4i)T + 8.52e11iT^{2} \)
37 \( 1 + (-6.39e5 + 6.39e5i)T - 3.51e12iT^{2} \)
41 \( 1 + (8.30e5 - 8.30e5i)T - 7.98e12iT^{2} \)
43 \( 1 + (3.90e6 + 3.90e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (1.21e6 - 1.21e6i)T - 2.38e13iT^{2} \)
53 \( 1 + 8.26e6T + 6.22e13T^{2} \)
59 \( 1 + 8.93e6T + 1.46e14T^{2} \)
61 \( 1 + (1.58e7 + 1.58e7i)T + 1.91e14iT^{2} \)
67 \( 1 - 8.41e6iT - 4.06e14T^{2} \)
71 \( 1 + 2.39e7iT - 6.45e14T^{2} \)
73 \( 1 + (1.42e7 - 1.42e7i)T - 8.06e14iT^{2} \)
79 \( 1 + (-1.40e7 - 1.40e7i)T + 1.51e15iT^{2} \)
83 \( 1 - 1.12e7T + 2.25e15T^{2} \)
89 \( 1 + (2.74e7 + 2.74e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (3.01e7 - 3.01e7i)T - 7.83e15iT^{2} \)
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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

−14.31260818788135935406601653542, −13.28405008767918666132418211419, −12.04426686954560133921817055941, −10.91466881076506961394659586099, −9.314680761386080709277070119321, −6.85289023445495161819864686904, −6.41942425744887491643928169914, −4.78169997214755412514188738991, −1.65835689229963666975431333346, −0.06479638303635497132922752378, 3.20290373713619865983182465791, 4.59901337602835574556344090185, 6.12242630839201595928071982337, 8.185432469362387155547015205461, 10.03463239415376462037933252508, 10.97324361355294947229561448395, 12.25584338157386060996613961541, 13.21482690777306057325990243102, 15.18604890844124140444870758630, 16.18384032875078830600488916043

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