Properties

Label 2-29-29.2-c8-0-7
Degree $2$
Conductor $29$
Sign $0.947 + 0.318i$
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  + (−7.14 − 0.804i)2-s + (−31.1 − 19.5i)3-s + (−199. − 45.4i)4-s + (−619. + 493. i)5-s + (206. + 164. i)6-s + (232. + 1.01e3i)7-s + (3.12e3 + 1.09e3i)8-s + (−2.26e3 − 4.69e3i)9-s + (4.82e3 − 3.02e3i)10-s + (−5.46e3 + 1.91e3i)11-s + (5.31e3 + 5.31e3i)12-s + (5.18e3 − 1.07e4i)13-s + (−840. − 7.46e3i)14-s + (2.89e4 − 3.26e3i)15-s + (2.57e4 + 1.23e4i)16-s + (8.34e4 − 8.34e4i)17-s + ⋯
L(s)  = 1  + (−0.446 − 0.0502i)2-s + (−0.384 − 0.241i)3-s + (−0.778 − 0.177i)4-s + (−0.991 + 0.790i)5-s + (0.159 + 0.127i)6-s + (0.0968 + 0.424i)7-s + (0.762 + 0.266i)8-s + (−0.344 − 0.715i)9-s + (0.482 − 0.302i)10-s + (−0.373 + 0.130i)11-s + (0.256 + 0.256i)12-s + (0.181 − 0.377i)13-s + (−0.0218 − 0.194i)14-s + (0.571 − 0.0643i)15-s + (0.392 + 0.188i)16-s + (0.998 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.947 + 0.318i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ 0.947 + 0.318i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.655536 - 0.107207i\)
\(L(\frac12)\) \(\approx\) \(0.655536 - 0.107207i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-3.03e5 - 6.38e5i)T \)
good2 \( 1 + (7.14 + 0.804i)T + (249. + 56.9i)T^{2} \)
3 \( 1 + (31.1 + 19.5i)T + (2.84e3 + 5.91e3i)T^{2} \)
5 \( 1 + (619. - 493. i)T + (8.69e4 - 3.80e5i)T^{2} \)
7 \( 1 + (-232. - 1.01e3i)T + (-5.19e6 + 2.50e6i)T^{2} \)
11 \( 1 + (5.46e3 - 1.91e3i)T + (1.67e8 - 1.33e8i)T^{2} \)
13 \( 1 + (-5.18e3 + 1.07e4i)T + (-5.08e8 - 6.37e8i)T^{2} \)
17 \( 1 + (-8.34e4 + 8.34e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (-5.14e4 - 8.19e4i)T + (-7.36e9 + 1.53e10i)T^{2} \)
23 \( 1 + (-9.57e4 + 1.20e5i)T + (-1.74e10 - 7.63e10i)T^{2} \)
31 \( 1 + (3.15e4 + 3.55e3i)T + (8.31e11 + 1.89e11i)T^{2} \)
37 \( 1 + (7.83e5 + 2.74e5i)T + (2.74e12 + 2.18e12i)T^{2} \)
41 \( 1 + (3.91e5 + 3.91e5i)T + 7.98e12iT^{2} \)
43 \( 1 + (-2.27e5 - 2.01e6i)T + (-1.13e13 + 2.60e12i)T^{2} \)
47 \( 1 + (1.78e6 + 5.11e6i)T + (-1.86e13 + 1.48e13i)T^{2} \)
53 \( 1 + (-7.21e5 - 9.04e5i)T + (-1.38e13 + 6.06e13i)T^{2} \)
59 \( 1 - 2.09e7T + 1.46e14T^{2} \)
61 \( 1 + (-1.51e7 - 9.51e6i)T + (8.31e13 + 1.72e14i)T^{2} \)
67 \( 1 + (5.79e6 + 1.20e7i)T + (-2.53e14 + 3.17e14i)T^{2} \)
71 \( 1 + (-7.79e6 + 1.61e7i)T + (-4.02e14 - 5.04e14i)T^{2} \)
73 \( 1 + (-3.24e7 + 3.65e6i)T + (7.86e14 - 1.79e14i)T^{2} \)
79 \( 1 + (3.05e5 - 8.72e5i)T + (-1.18e15 - 9.45e14i)T^{2} \)
83 \( 1 + (2.99e6 - 1.31e7i)T + (-2.02e15 - 9.77e14i)T^{2} \)
89 \( 1 + (1.11e8 + 1.25e7i)T + (3.83e15 + 8.75e14i)T^{2} \)
97 \( 1 + (-1.81e7 + 1.13e7i)T + (3.40e15 - 7.06e15i)T^{2} \)
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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

−15.14999889996524787728258668147, −14.19397623063630280635495486797, −12.48884565287858557449594380606, −11.40043422131824602267985440236, −10.06123892288575912133291324607, −8.549480581451411260872847916520, −7.24994038789128139179632811026, −5.37184262628403123016564710984, −3.39435956731348691595444154278, −0.63877766896018814920169217877, 0.75901844856468704805772101870, 3.99079700116303508491098835277, 5.17679180869717235627916811801, 7.71902368398281697985257185712, 8.556771682600381212692914721269, 10.13202430624715326822007828901, 11.47595609940155993645971321168, 12.83827121471595330097159843056, 14.00065312124827111943338146107, 15.76397833266398224776133965797

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