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Properties

Label 2-29-29.17-c6-0-12
Degree $2$
Conductor $29$
Sign $-0.993 + 0.116i$
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  + (8.70 − 8.70i)2-s + (−29.1 + 29.1i)3-s − 87.6i·4-s − 207. i·5-s + 507. i·6-s − 652.·7-s + (−205. − 205. i)8-s − 972. i·9-s + (−1.80e3 − 1.80e3i)10-s + (9.99 − 9.99i)11-s + (2.55e3 + 2.55e3i)12-s − 1.29e3i·13-s + (−5.68e3 + 5.68e3i)14-s + (6.03e3 + 6.03e3i)15-s + 2.02e3·16-s + (351. − 351. i)17-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)2-s + (−1.08 + 1.08i)3-s − 1.36i·4-s − 1.65i·5-s + 2.35i·6-s − 1.90·7-s + (−0.401 − 0.401i)8-s − 1.33i·9-s + (−1.80 − 1.80i)10-s + (0.00750 − 0.00750i)11-s + (1.47 + 1.47i)12-s − 0.589i·13-s + (−2.07 + 2.07i)14-s + (1.78 + 1.78i)15-s + 0.494·16-s + (0.0715 − 0.0715i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.993 + 0.116i$
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: \((2,\ 29,\ (\ :3),\ -0.993 + 0.116i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0625886 - 1.06875i\)
\(L(\frac12)\) \(\approx\) \(0.0625886 - 1.06875i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (2.32e4 - 7.37e3i)T \)
good2 \( 1 + (-8.70 + 8.70i)T - 64iT^{2} \)
3 \( 1 + (29.1 - 29.1i)T - 729iT^{2} \)
5 \( 1 + 207. iT - 1.56e4T^{2} \)
7 \( 1 + 652.T + 1.17e5T^{2} \)
11 \( 1 + (-9.99 + 9.99i)T - 1.77e6iT^{2} \)
13 \( 1 + 1.29e3iT - 4.82e6T^{2} \)
17 \( 1 + (-351. + 351. i)T - 2.41e7iT^{2} \)
19 \( 1 + (-1.59e3 + 1.59e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 1.02e4T + 1.48e8T^{2} \)
31 \( 1 + (-2.41e4 + 2.41e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (4.43e4 + 4.43e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (4.69e3 + 4.69e3i)T + 4.75e9iT^{2} \)
43 \( 1 + (7.26e4 - 7.26e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (3.40e4 + 3.40e4i)T + 1.07e10iT^{2} \)
53 \( 1 + 6.64e3T + 2.21e10T^{2} \)
59 \( 1 + 2.65e4T + 4.21e10T^{2} \)
61 \( 1 + (-1.94e5 + 1.94e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 3.01e4iT - 9.04e10T^{2} \)
71 \( 1 - 1.42e5iT - 1.28e11T^{2} \)
73 \( 1 + (5.39e5 + 5.39e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (1.33e5 - 1.33e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 2.08e5T + 3.26e11T^{2} \)
89 \( 1 + (-1.32e5 + 1.32e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (-3.04e5 - 3.04e5i)T + 8.32e11iT^{2} \)
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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

−15.46792049460580762361429843625, −13.22959878506405929587450157170, −12.70217016615406199474687007815, −11.65759378506571204741487243393, −10.24466990695119207248367889720, −9.329452988829450132422394967138, −5.82163921669484863077793355173, −4.84677349120497344284074155564, −3.54309548268567203397210410645, −0.46583702487634873953191132316, 3.29236422471857148883018078201, 5.84954367216966411452846496900, 6.75142430562453447126506085054, 7.07029630167192169095019475306, 10.23729340488129591881270031439, 11.82422038742836944337775331092, 12.98457530649788094105783838359, 13.81230460710076579858310156840, 15.16163148563328646099258407023, 16.26355891425520739745782480526

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