Properties

Label 2-29-29.17-c6-0-1
Degree $2$
Conductor $29$
Sign $-0.520 + 0.853i$
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  + (−6.25 + 6.25i)2-s + (−11.2 + 11.2i)3-s − 14.2i·4-s + 192. i·5-s − 140. i·6-s − 54.6·7-s + (−311. − 311. i)8-s + 477. i·9-s + (−1.20e3 − 1.20e3i)10-s + (785. − 785. i)11-s + (159. + 159. i)12-s − 3.66e3i·13-s + (341. − 341. i)14-s + (−2.16e3 − 2.16e3i)15-s + 4.80e3·16-s + (−4.65e3 + 4.65e3i)17-s + ⋯
L(s)  = 1  + (−0.781 + 0.781i)2-s + (−0.415 + 0.415i)3-s − 0.222i·4-s + 1.54i·5-s − 0.649i·6-s − 0.159·7-s + (−0.607 − 0.607i)8-s + 0.655i·9-s + (−1.20 − 1.20i)10-s + (0.590 − 0.590i)11-s + (0.0923 + 0.0923i)12-s − 1.66i·13-s + (0.124 − 0.124i)14-s + (−0.640 − 0.640i)15-s + 1.17·16-s + (−0.947 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.520 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.520 + 0.853i$
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: \((2,\ 29,\ (\ :3),\ -0.520 + 0.853i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.236019 - 0.420518i\)
\(L(\frac12)\) \(\approx\) \(0.236019 - 0.420518i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (8.53e3 - 2.28e4i)T \)
good2 \( 1 + (6.25 - 6.25i)T - 64iT^{2} \)
3 \( 1 + (11.2 - 11.2i)T - 729iT^{2} \)
5 \( 1 - 192. iT - 1.56e4T^{2} \)
7 \( 1 + 54.6T + 1.17e5T^{2} \)
11 \( 1 + (-785. + 785. i)T - 1.77e6iT^{2} \)
13 \( 1 + 3.66e3iT - 4.82e6T^{2} \)
17 \( 1 + (4.65e3 - 4.65e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (4.78e3 - 4.78e3i)T - 4.70e7iT^{2} \)
23 \( 1 - 2.03e4T + 1.48e8T^{2} \)
31 \( 1 + (5.99e3 - 5.99e3i)T - 8.87e8iT^{2} \)
37 \( 1 + (1.39e4 + 1.39e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (3.25e4 + 3.25e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (3.92e4 - 3.92e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-7.76e3 - 7.76e3i)T + 1.07e10iT^{2} \)
53 \( 1 - 1.97e5T + 2.21e10T^{2} \)
59 \( 1 + 2.92e5T + 4.21e10T^{2} \)
61 \( 1 + (9.47e4 - 9.47e4i)T - 5.15e10iT^{2} \)
67 \( 1 - 1.47e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.18e5iT - 1.28e11T^{2} \)
73 \( 1 + (1.79e5 + 1.79e5i)T + 1.51e11iT^{2} \)
79 \( 1 + (3.23e5 - 3.23e5i)T - 2.43e11iT^{2} \)
83 \( 1 - 2.78e5T + 3.26e11T^{2} \)
89 \( 1 + (-3.97e5 + 3.97e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (2.54e5 + 2.54e5i)T + 8.32e11iT^{2} \)
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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

−16.75985190996216091730142121833, −15.47397710749433615361583614898, −14.75911180010395275746229895297, −12.97321343092038386409535370897, −10.98763509134804589350562442705, −10.32052205812473088617612958047, −8.520950555046526628950929400593, −7.15469730148552357237854043568, −5.96565611089486158442711433592, −3.28995355296540714478465797924, 0.36429902815028652294938213427, 1.68189550912806283903797966457, 4.70362781175108423110305018721, 6.66462791840985011573401707996, 9.070653355368619917541639132922, 9.248094828006066964386743220246, 11.39913291703435746492975358997, 12.08930304250154002831100696617, 13.31158313420934487603413085278, 15.13265568174024776650719238757

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