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Properties

Label 2-29-29.12-c6-0-11
Degree $2$
Conductor $29$
Sign $-0.695 + 0.718i$
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  + (−0.624 − 0.624i)2-s + (1.83 + 1.83i)3-s − 63.2i·4-s + 41.8i·5-s − 2.29i·6-s − 304.·7-s + (−79.4 + 79.4i)8-s − 722. i·9-s + (26.0 − 26.0i)10-s + (−1.22e3 − 1.22e3i)11-s + (116. − 116. i)12-s − 606. i·13-s + (189. + 189. i)14-s + (−76.8 + 76.8i)15-s − 3.94e3·16-s + (−4.14e3 − 4.14e3i)17-s + ⋯
L(s)  = 1  + (−0.0780 − 0.0780i)2-s + (0.0680 + 0.0680i)3-s − 0.987i·4-s + 0.334i·5-s − 0.0106i·6-s − 0.886·7-s + (−0.155 + 0.155i)8-s − 0.990i·9-s + (0.0260 − 0.0260i)10-s + (−0.920 − 0.920i)11-s + (0.0672 − 0.0672i)12-s − 0.276i·13-s + (0.0691 + 0.0691i)14-s + (−0.0227 + 0.0227i)15-s − 0.963·16-s + (−0.844 − 0.844i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.695 + 0.718i$
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: \((2,\ 29,\ (\ :3),\ -0.695 + 0.718i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.362777 - 0.856011i\)
\(L(\frac12)\) \(\approx\) \(0.362777 - 0.856011i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (1.99e4 - 1.40e4i)T \)
good2 \( 1 + (0.624 + 0.624i)T + 64iT^{2} \)
3 \( 1 + (-1.83 - 1.83i)T + 729iT^{2} \)
5 \( 1 - 41.8iT - 1.56e4T^{2} \)
7 \( 1 + 304.T + 1.17e5T^{2} \)
11 \( 1 + (1.22e3 + 1.22e3i)T + 1.77e6iT^{2} \)
13 \( 1 + 606. iT - 4.82e6T^{2} \)
17 \( 1 + (4.14e3 + 4.14e3i)T + 2.41e7iT^{2} \)
19 \( 1 + (-6.48e3 - 6.48e3i)T + 4.70e7iT^{2} \)
23 \( 1 - 9.53e3T + 1.48e8T^{2} \)
31 \( 1 + (-1.38e4 - 1.38e4i)T + 8.87e8iT^{2} \)
37 \( 1 + (-5.72e4 + 5.72e4i)T - 2.56e9iT^{2} \)
41 \( 1 + (3.63e3 - 3.63e3i)T - 4.75e9iT^{2} \)
43 \( 1 + (8.71e4 + 8.71e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (-4.02e4 + 4.02e4i)T - 1.07e10iT^{2} \)
53 \( 1 + 9.62e4T + 2.21e10T^{2} \)
59 \( 1 - 3.11e5T + 4.21e10T^{2} \)
61 \( 1 + (-3.39e4 - 3.39e4i)T + 5.15e10iT^{2} \)
67 \( 1 + 2.31e5iT - 9.04e10T^{2} \)
71 \( 1 - 3.58e5iT - 1.28e11T^{2} \)
73 \( 1 + (-3.37e5 + 3.37e5i)T - 1.51e11iT^{2} \)
79 \( 1 + (1.72e5 + 1.72e5i)T + 2.43e11iT^{2} \)
83 \( 1 + 6.06e5T + 3.26e11T^{2} \)
89 \( 1 + (6.97e5 + 6.97e5i)T + 4.96e11iT^{2} \)
97 \( 1 + (3.88e4 - 3.88e4i)T - 8.32e11iT^{2} \)
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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

−15.37759200357423503816728935951, −14.21842975768034749672292550179, −12.98242120162900321070253649334, −11.29871453376149723221543061166, −10.11268200398735566824458295226, −9.010067026771223828129744631944, −6.83292974631985938970009723333, −5.52357270554568524026722300162, −3.09602190048261842954152983417, −0.50680235123265465801893920515, 2.67527185956787632847104893887, 4.68970471834733232303948160897, 6.94461486394659019627887736332, 8.184266403936213889412785065963, 9.625881231224112739747516754093, 11.28139782805451539331873661066, 12.96483119818824536295006618008, 13.20569967080303555950141487636, 15.36428312448791420079337076088, 16.33923994942013292851964863792

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