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Properties

Label 2-29-29.12-c8-0-1
Degree $2$
Conductor $29$
Sign $0.527 + 0.849i$
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  + (−12.8 − 12.8i)2-s + (−84.9 − 84.9i)3-s + 74.6i·4-s + 733. i·5-s + 2.18e3i·6-s + 4.37e3·7-s + (−2.33e3 + 2.33e3i)8-s + 7.86e3i·9-s + (9.42e3 − 9.42e3i)10-s + (−1.01e4 − 1.01e4i)11-s + (6.33e3 − 6.33e3i)12-s + 3.12e4i·13-s + (−5.62e4 − 5.62e4i)14-s + (6.23e4 − 6.23e4i)15-s + 7.90e4·16-s + (−1.08e4 − 1.08e4i)17-s + ⋯
L(s)  = 1  + (−0.803 − 0.803i)2-s + (−1.04 − 1.04i)3-s + 0.291i·4-s + 1.17i·5-s + 1.68i·6-s + 1.82·7-s + (−0.569 + 0.569i)8-s + 1.19i·9-s + (0.942 − 0.942i)10-s + (−0.694 − 0.694i)11-s + (0.305 − 0.305i)12-s + 1.09i·13-s + (−1.46 − 1.46i)14-s + (1.23 − 1.23i)15-s + 1.20·16-s + (−0.130 − 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.527 + 0.849i$
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: \((2,\ 29,\ (\ :4),\ 0.527 + 0.849i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.734522 - 0.408427i\)
\(L(\frac12)\) \(\approx\) \(0.734522 - 0.408427i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.52e5 + 6.60e5i)T \)
good2 \( 1 + (12.8 + 12.8i)T + 256iT^{2} \)
3 \( 1 + (84.9 + 84.9i)T + 6.56e3iT^{2} \)
5 \( 1 - 733. iT - 3.90e5T^{2} \)
7 \( 1 - 4.37e3T + 5.76e6T^{2} \)
11 \( 1 + (1.01e4 + 1.01e4i)T + 2.14e8iT^{2} \)
13 \( 1 - 3.12e4iT - 8.15e8T^{2} \)
17 \( 1 + (1.08e4 + 1.08e4i)T + 6.97e9iT^{2} \)
19 \( 1 + (-4.04e4 - 4.04e4i)T + 1.69e10iT^{2} \)
23 \( 1 - 2.66e5T + 7.83e10T^{2} \)
31 \( 1 + (-9.54e5 - 9.54e5i)T + 8.52e11iT^{2} \)
37 \( 1 + (-7.19e5 + 7.19e5i)T - 3.51e12iT^{2} \)
41 \( 1 + (2.14e6 - 2.14e6i)T - 7.98e12iT^{2} \)
43 \( 1 + (1.84e6 + 1.84e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-5.63e6 + 5.63e6i)T - 2.38e13iT^{2} \)
53 \( 1 - 1.75e6T + 6.22e13T^{2} \)
59 \( 1 - 1.63e7T + 1.46e14T^{2} \)
61 \( 1 + (-1.21e7 - 1.21e7i)T + 1.91e14iT^{2} \)
67 \( 1 + 7.31e5iT - 4.06e14T^{2} \)
71 \( 1 - 1.93e7iT - 6.45e14T^{2} \)
73 \( 1 + (-2.46e6 + 2.46e6i)T - 8.06e14iT^{2} \)
79 \( 1 + (-2.38e7 - 2.38e7i)T + 1.51e15iT^{2} \)
83 \( 1 + 9.11e7T + 2.25e15T^{2} \)
89 \( 1 + (-6.67e7 - 6.67e7i)T + 3.93e15iT^{2} \)
97 \( 1 + (1.00e8 - 1.00e8i)T - 7.83e15iT^{2} \)
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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

−14.89125848364018358473224824467, −13.79484602991169167954459954486, −11.75181758156010819498193675034, −11.35575119092831479079382606471, −10.49605137319333064144212817901, −8.366026300179674126877394186215, −6.99432106767148105557109367955, −5.42702206635743380820195085530, −2.25886944133592057223429789803, −1.00971768724833523373185972404, 0.76095950002294628506052460202, 4.67782006435945748147769497793, 5.41469985028778055755723980905, 7.70414326956264622720149919984, 8.756214141795267447274096108970, 10.18394844253182533569191792224, 11.44753181283333758863951534966, 12.75932983669243699516608396204, 15.06759372432626687108470554604, 15.71321632155202501036700188133

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