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Properties

Label 2-29-29.28-c5-0-7
Degree $2$
Conductor $29$
Sign $0.912 - 0.409i$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.02i·2-s − 26.5i·3-s − 49.4·4-s + 71.1·5-s + 239.·6-s + 131.·7-s − 157. i·8-s − 463.·9-s + 641. i·10-s − 522. i·11-s + 1.31e3i·12-s + 855.·13-s + 1.18e3i·14-s − 1.88e3i·15-s − 163.·16-s + 1.26e3i·17-s + ⋯
L(s)  = 1  + 1.59i·2-s − 1.70i·3-s − 1.54·4-s + 1.27·5-s + 2.71·6-s + 1.01·7-s − 0.867i·8-s − 1.90·9-s + 2.02i·10-s − 1.30i·11-s + 2.63i·12-s + 1.40·13-s + 1.62i·14-s − 2.16i·15-s − 0.160·16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.912 - 0.409i$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ 0.912 - 0.409i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.75267 + 0.374935i\)
\(L(\frac12)\) \(\approx\) \(1.75267 + 0.374935i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (4.13e3 - 1.85e3i)T \)
good2 \( 1 - 9.02iT - 32T^{2} \)
3 \( 1 + 26.5iT - 243T^{2} \)
5 \( 1 - 71.1T + 3.12e3T^{2} \)
7 \( 1 - 131.T + 1.68e4T^{2} \)
11 \( 1 + 522. iT - 1.61e5T^{2} \)
13 \( 1 - 855.T + 3.71e5T^{2} \)
17 \( 1 - 1.26e3iT - 1.41e6T^{2} \)
19 \( 1 + 73.8iT - 2.47e6T^{2} \)
23 \( 1 + 2.40e3T + 6.43e6T^{2} \)
31 \( 1 + 2.05e3iT - 2.86e7T^{2} \)
37 \( 1 - 8.66e3iT - 6.93e7T^{2} \)
41 \( 1 - 5.91e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.80e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.11e4iT - 2.29e8T^{2} \)
53 \( 1 + 1.44e3T + 4.18e8T^{2} \)
59 \( 1 - 2.99e4T + 7.14e8T^{2} \)
61 \( 1 - 3.68e3iT - 8.44e8T^{2} \)
67 \( 1 - 978.T + 1.35e9T^{2} \)
71 \( 1 + 8.00e4T + 1.80e9T^{2} \)
73 \( 1 + 2.09e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.42e4iT - 3.07e9T^{2} \)
83 \( 1 - 9.42e4T + 3.93e9T^{2} \)
89 \( 1 - 7.22e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.53e5iT - 8.58e9T^{2} \)
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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

−16.44805438732666327835587759553, −14.61811639653798450084868690895, −13.73808536407229209226983365949, −13.23388604625367177643602441605, −11.24185282967945567642267870954, −8.651160603316658846233164008709, −7.929580118872140590645560706197, −6.25685510736776458181844512471, −5.82817497466360301185762848595, −1.53166218646922556399458790035, 2.02880962912196080507506164948, 3.99219231662652460473409827594, 5.26954212616191211933957061471, 8.999232976454900130677962315969, 9.866538477290266221444942180787, 10.66575969784888133520570781521, 11.70929907554374994049480387450, 13.48771557724226487835222385448, 14.53902158369048600907252862326, 15.96475190685598398986150180585

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