Properties

Label 2-29-1.1-c5-0-6
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $4.65113$
Root an. cond. $2.15664$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.83·2-s + 15.9·3-s + 2.10·4-s + 31.5·5-s + 93.2·6-s + 106.·7-s − 174.·8-s + 11.9·9-s + 184.·10-s − 152.·11-s + 33.5·12-s + 325.·13-s + 620.·14-s + 503.·15-s − 1.08e3·16-s − 664.·17-s + 69.9·18-s − 1.59e3·19-s + 66.3·20-s + 1.69e3·21-s − 892.·22-s + 719.·23-s − 2.78e3·24-s − 2.12e3·25-s + 1.90e3·26-s − 3.68e3·27-s + 223.·28-s + ⋯
L(s)  = 1  + 1.03·2-s + 1.02·3-s + 0.0656·4-s + 0.564·5-s + 1.05·6-s + 0.819·7-s − 0.964·8-s + 0.0492·9-s + 0.582·10-s − 0.380·11-s + 0.0672·12-s + 0.534·13-s + 0.846·14-s + 0.578·15-s − 1.06·16-s − 0.558·17-s + 0.0508·18-s − 1.01·19-s + 0.0370·20-s + 0.839·21-s − 0.393·22-s + 0.283·23-s − 0.988·24-s − 0.681·25-s + 0.551·26-s − 0.973·27-s + 0.0538·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(4.65113\)
Root analytic conductor: \(2.15664\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.085435827\)
\(L(\frac12)\) \(\approx\) \(3.085435827\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 841T \)
good2 \( 1 - 5.83T + 32T^{2} \)
3 \( 1 - 15.9T + 243T^{2} \)
5 \( 1 - 31.5T + 3.12e3T^{2} \)
7 \( 1 - 106.T + 1.68e4T^{2} \)
11 \( 1 + 152.T + 1.61e5T^{2} \)
13 \( 1 - 325.T + 3.71e5T^{2} \)
17 \( 1 + 664.T + 1.41e6T^{2} \)
19 \( 1 + 1.59e3T + 2.47e6T^{2} \)
23 \( 1 - 719.T + 6.43e6T^{2} \)
31 \( 1 - 2.05e3T + 2.86e7T^{2} \)
37 \( 1 - 1.49e4T + 6.93e7T^{2} \)
41 \( 1 - 1.46e4T + 1.15e8T^{2} \)
43 \( 1 - 1.02e4T + 1.47e8T^{2} \)
47 \( 1 - 4.58e3T + 2.29e8T^{2} \)
53 \( 1 - 8.95e3T + 4.18e8T^{2} \)
59 \( 1 + 1.37e4T + 7.14e8T^{2} \)
61 \( 1 + 3.34e4T + 8.44e8T^{2} \)
67 \( 1 - 3.75e4T + 1.35e9T^{2} \)
71 \( 1 + 1.47e4T + 1.80e9T^{2} \)
73 \( 1 - 6.32e4T + 2.07e9T^{2} \)
79 \( 1 + 2.71e4T + 3.07e9T^{2} \)
83 \( 1 + 5.44e4T + 3.93e9T^{2} \)
89 \( 1 + 1.39e5T + 5.58e9T^{2} \)
97 \( 1 - 8.98e4T + 8.58e9T^{2} \)
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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

−15.45086268707811700101389335878, −14.52955262388493593573834313838, −13.76101987130523354169576363226, −12.87103024720905609629137865900, −11.18267074703925794887013899301, −9.291424804636731510457707136658, −8.163271023559568898557066072591, −5.93300357995893759225022693796, −4.29057070580289617996936758336, −2.49957508958904923537462330040, 2.49957508958904923537462330040, 4.29057070580289617996936758336, 5.93300357995893759225022693796, 8.163271023559568898557066072591, 9.291424804636731510457707136658, 11.18267074703925794887013899301, 12.87103024720905609629137865900, 13.76101987130523354169576363226, 14.52955262388493593573834313838, 15.45086268707811700101389335878

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