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Properties

Label 2-29-29.17-c8-0-9
Degree $2$
Conductor $29$
Sign $-0.750 - 0.660i$
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  + (−21.4 + 21.4i)2-s + (−26.9 + 26.9i)3-s − 667. i·4-s + 209. i·5-s − 1.15e3i·6-s + 2.37e3·7-s + (8.85e3 + 8.85e3i)8-s + 5.10e3i·9-s + (−4.51e3 − 4.51e3i)10-s + (1.64e4 − 1.64e4i)11-s + (1.80e4 + 1.80e4i)12-s − 722. i·13-s + (−5.09e4 + 5.09e4i)14-s + (−5.66e3 − 5.66e3i)15-s − 2.09e5·16-s + (4.29e4 − 4.29e4i)17-s + ⋯
L(s)  = 1  + (−1.34 + 1.34i)2-s + (−0.333 + 0.333i)3-s − 2.60i·4-s + 0.335i·5-s − 0.894i·6-s + 0.987·7-s + (2.16 + 2.16i)8-s + 0.778i·9-s + (−0.451 − 0.451i)10-s + (1.12 − 1.12i)11-s + (0.868 + 0.868i)12-s − 0.0253i·13-s + (−1.32 + 1.32i)14-s + (−0.111 − 0.111i)15-s − 3.19·16-s + (0.513 − 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.750 - 0.660i$
Analytic conductor: \(11.8139\)
Root analytic conductor: \(3.43714\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :4),\ -0.750 - 0.660i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.302161 + 0.801027i\)
\(L(\frac12)\) \(\approx\) \(0.302161 + 0.801027i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-6.09e5 - 3.58e5i)T \)
good2 \( 1 + (21.4 - 21.4i)T - 256iT^{2} \)
3 \( 1 + (26.9 - 26.9i)T - 6.56e3iT^{2} \)
5 \( 1 - 209. iT - 3.90e5T^{2} \)
7 \( 1 - 2.37e3T + 5.76e6T^{2} \)
11 \( 1 + (-1.64e4 + 1.64e4i)T - 2.14e8iT^{2} \)
13 \( 1 + 722. iT - 8.15e8T^{2} \)
17 \( 1 + (-4.29e4 + 4.29e4i)T - 6.97e9iT^{2} \)
19 \( 1 + (1.02e5 - 1.02e5i)T - 1.69e10iT^{2} \)
23 \( 1 - 4.53e4T + 7.83e10T^{2} \)
31 \( 1 + (1.03e6 - 1.03e6i)T - 8.52e11iT^{2} \)
37 \( 1 + (-7.42e5 - 7.42e5i)T + 3.51e12iT^{2} \)
41 \( 1 + (-2.06e6 - 2.06e6i)T + 7.98e12iT^{2} \)
43 \( 1 + (2.46e6 - 2.46e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (-1.62e6 - 1.62e6i)T + 2.38e13iT^{2} \)
53 \( 1 - 5.35e5T + 6.22e13T^{2} \)
59 \( 1 - 1.94e7T + 1.46e14T^{2} \)
61 \( 1 + (1.18e7 - 1.18e7i)T - 1.91e14iT^{2} \)
67 \( 1 + 3.65e7iT - 4.06e14T^{2} \)
71 \( 1 - 1.99e7iT - 6.45e14T^{2} \)
73 \( 1 + (-9.33e6 - 9.33e6i)T + 8.06e14iT^{2} \)
79 \( 1 + (-5.09e7 + 5.09e7i)T - 1.51e15iT^{2} \)
83 \( 1 + 2.13e7T + 2.25e15T^{2} \)
89 \( 1 + (1.63e7 - 1.63e7i)T - 3.93e15iT^{2} \)
97 \( 1 + (-4.08e7 - 4.08e7i)T + 7.83e15iT^{2} \)
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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

−16.31693658139460471394071431168, −14.71471594385475144153063085977, −14.16039732690036504987316687680, −11.26902193456362187195930685222, −10.42568985675329025349915030209, −8.875947077168307459928602325839, −7.87354800401649048149740835764, −6.40991811626786631213596700062, −5.06315367764141320500483403525, −1.24135646849738451612334984544, 0.76056740123295008904927555164, 1.92017475797037800751168980692, 4.11851756731894618338803974230, 7.09191874499283669836938205406, 8.574758244282798313268071934674, 9.587668607697797160286967871824, 11.03707077848214501256073512339, 11.99698807380927996891187674049, 12.75925201471255116405091654060, 14.85696910371103382403771285105

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