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Properties

Label 2-29-29.28-c9-0-15
Degree $2$
Conductor $29$
Sign $0.834 + 0.551i$
Analytic cond. $14.9360$
Root an. cond. $3.86471$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.11i·2-s − 171. i·3-s + 428.·4-s + 2.21e3·5-s + 1.56e3·6-s − 1.87e3·7-s + 8.57e3i·8-s − 9.66e3·9-s + 2.02e4i·10-s − 7.04e3i·11-s − 7.34e4i·12-s + 4.25e4·13-s − 1.70e4i·14-s − 3.79e5i·15-s + 1.41e5·16-s + 8.82e4i·17-s + ⋯
L(s)  = 1  + 0.402i·2-s − 1.22i·3-s + 0.837·4-s + 1.58·5-s + 0.491·6-s − 0.294·7-s + 0.740i·8-s − 0.491·9-s + 0.639i·10-s − 0.145i·11-s − 1.02i·12-s + 0.413·13-s − 0.118i·14-s − 1.93i·15-s + 0.539·16-s + 0.256i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.834 + 0.551i$
Analytic conductor: \(14.9360\)
Root analytic conductor: \(3.86471\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :9/2),\ 0.834 + 0.551i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.77799 - 0.834744i\)
\(L(\frac12)\) \(\approx\) \(2.77799 - 0.834744i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.17e6 + 2.09e6i)T \)
good2 \( 1 - 9.11iT - 512T^{2} \)
3 \( 1 + 171. iT - 1.96e4T^{2} \)
5 \( 1 - 2.21e3T + 1.95e6T^{2} \)
7 \( 1 + 1.87e3T + 4.03e7T^{2} \)
11 \( 1 + 7.04e3iT - 2.35e9T^{2} \)
13 \( 1 - 4.25e4T + 1.06e10T^{2} \)
17 \( 1 - 8.82e4iT - 1.18e11T^{2} \)
19 \( 1 - 8.13e3iT - 3.22e11T^{2} \)
23 \( 1 - 1.17e5T + 1.80e12T^{2} \)
31 \( 1 + 7.39e6iT - 2.64e13T^{2} \)
37 \( 1 + 1.28e7iT - 1.29e14T^{2} \)
41 \( 1 - 2.51e7iT - 3.27e14T^{2} \)
43 \( 1 - 3.64e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.99e6iT - 1.11e15T^{2} \)
53 \( 1 + 5.45e7T + 3.29e15T^{2} \)
59 \( 1 + 2.16e7T + 8.66e15T^{2} \)
61 \( 1 - 1.42e8iT - 1.16e16T^{2} \)
67 \( 1 + 7.07e7T + 2.72e16T^{2} \)
71 \( 1 - 3.47e8T + 4.58e16T^{2} \)
73 \( 1 - 2.96e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.79e8iT - 1.19e17T^{2} \)
83 \( 1 - 1.33e8T + 1.86e17T^{2} \)
89 \( 1 - 8.73e8iT - 3.50e17T^{2} \)
97 \( 1 - 2.02e8iT - 7.60e17T^{2} \)
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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

−14.75689574127516586856441898124, −13.54197817446912750703552941916, −12.77028136339681507025243863136, −11.21403715809591425893789318298, −9.670364131511013394481118144022, −7.86562637124223049548885148126, −6.50422147917986516604702678152, −5.86309448562211213920868869148, −2.44765147065070835585830514520, −1.40016606482689178365865961465, 1.72976878940778043466649949453, 3.30559006318679187069355502792, 5.29395895214653249308482648899, 6.67884614368429623812253527581, 9.203576618580686192796099604825, 10.11964424000728236015466083790, 10.88972075625661426479010924148, 12.61775933262828210946941625622, 13.98803083246700169258885603436, 15.33476003046265915569763165156

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