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Properties

Label 2-29-29.17-c6-0-7
Degree $2$
Conductor $29$
Sign $0.966 - 0.258i$
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  + (−9.26 + 9.26i)2-s + (−17.8 + 17.8i)3-s − 107. i·4-s − 107. i·5-s − 331. i·6-s − 104.·7-s + (406. + 406. i)8-s + 91.4i·9-s + (1.00e3 + 1.00e3i)10-s + (264. − 264. i)11-s + (1.92e3 + 1.92e3i)12-s + 392. i·13-s + (964. − 964. i)14-s + (1.92e3 + 1.92e3i)15-s − 634.·16-s + (504. − 504. i)17-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)2-s + (−0.661 + 0.661i)3-s − 1.68i·4-s − 0.863i·5-s − 1.53i·6-s − 0.303·7-s + (0.794 + 0.794i)8-s + 0.125i·9-s + (1.00 + 1.00i)10-s + (0.198 − 0.198i)11-s + (1.11 + 1.11i)12-s + 0.178i·13-s + (0.351 − 0.351i)14-s + (0.571 + 0.571i)15-s − 0.154·16-s + (0.102 − 0.102i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.966 - 0.258i$
Analytic conductor: \(6.67156\)
Root analytic conductor: \(2.58293\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :3),\ 0.966 - 0.258i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.519278 + 0.0682495i\)
\(L(\frac12)\) \(\approx\) \(0.519278 + 0.0682495i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-2.19e4 + 1.06e4i)T \)
good2 \( 1 + (9.26 - 9.26i)T - 64iT^{2} \)
3 \( 1 + (17.8 - 17.8i)T - 729iT^{2} \)
5 \( 1 + 107. iT - 1.56e4T^{2} \)
7 \( 1 + 104.T + 1.17e5T^{2} \)
11 \( 1 + (-264. + 264. i)T - 1.77e6iT^{2} \)
13 \( 1 - 392. iT - 4.82e6T^{2} \)
17 \( 1 + (-504. + 504. i)T - 2.41e7iT^{2} \)
19 \( 1 + (-7.16e3 + 7.16e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.19e4T + 1.48e8T^{2} \)
31 \( 1 + (-3.57e4 + 3.57e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (-3.70e4 - 3.70e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (2.34e4 + 2.34e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (-7.93e4 + 7.93e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (9.51e4 + 9.51e4i)T + 1.07e10iT^{2} \)
53 \( 1 - 4.17e4T + 2.21e10T^{2} \)
59 \( 1 + 3.27e5T + 4.21e10T^{2} \)
61 \( 1 + (3.29e4 - 3.29e4i)T - 5.15e10iT^{2} \)
67 \( 1 + 1.46e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.31e4iT - 1.28e11T^{2} \)
73 \( 1 + (1.14e5 + 1.14e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (-1.18e5 + 1.18e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 6.13e5T + 3.26e11T^{2} \)
89 \( 1 + (-9.57e5 + 9.57e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-7.01e5 - 7.01e5i)T + 8.32e11iT^{2} \)
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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

−16.16160718842170129237744768809, −15.43314984431523204948310790700, −13.66712262548791160537412981400, −11.77386316866132859342556094702, −10.17466205997381746555805772022, −9.208318752140406675938154548372, −7.909134301495773690932141867309, −6.22571415675832464925607055445, −4.86907009870206282321816628329, −0.54866476482591657657089041990, 1.20802168593225934000035492366, 3.11900997914745353919075225928, 6.35133223814044936566364662836, 7.83252192394056354845121670216, 9.588891720244333078420105384548, 10.62883233489157127955173870368, 11.79637935029793870834028226318, 12.54582577446231102100128617304, 14.37821545516887424873434720744, 16.25745177146737778705382142174

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