Properties

Label 2-29-1.1-c17-0-4
Degree $2$
Conductor $29$
Sign $1$
Analytic cond. $53.1344$
Root an. cond. $7.28933$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 332.·2-s − 1.82e4·3-s − 2.02e4·4-s − 3.18e5·5-s + 6.06e6·6-s + 9.82e6·7-s + 5.03e7·8-s + 2.03e8·9-s + 1.06e8·10-s + 9.36e6·11-s + 3.69e8·12-s − 3.20e9·13-s − 3.26e9·14-s + 5.80e9·15-s − 1.41e10·16-s + 8.46e9·17-s − 6.75e10·18-s − 7.69e10·19-s + 6.46e9·20-s − 1.78e11·21-s − 3.11e9·22-s + 1.60e11·23-s − 9.18e11·24-s − 6.61e11·25-s + 1.06e12·26-s − 1.34e12·27-s − 1.99e11·28-s + ⋯
L(s)  = 1  − 0.919·2-s − 1.60·3-s − 0.154·4-s − 0.364·5-s + 1.47·6-s + 0.643·7-s + 1.06·8-s + 1.57·9-s + 0.335·10-s + 0.0131·11-s + 0.248·12-s − 1.09·13-s − 0.591·14-s + 0.584·15-s − 0.821·16-s + 0.294·17-s − 1.44·18-s − 1.03·19-s + 0.0564·20-s − 1.03·21-s − 0.0121·22-s + 0.428·23-s − 1.70·24-s − 0.867·25-s + 1.00·26-s − 0.917·27-s − 0.0996·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $1$
Analytic conductor: \(53.1344\)
Root analytic conductor: \(7.28933\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :17/2),\ 1)\)

Particular Values

\(L(9)\) \(\approx\) \(0.2164083308\)
\(L(\frac12)\) \(\approx\) \(0.2164083308\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 5.00e11T \)
good2 \( 1 + 332.T + 1.31e5T^{2} \)
3 \( 1 + 1.82e4T + 1.29e8T^{2} \)
5 \( 1 + 3.18e5T + 7.62e11T^{2} \)
7 \( 1 - 9.82e6T + 2.32e14T^{2} \)
11 \( 1 - 9.36e6T + 5.05e17T^{2} \)
13 \( 1 + 3.20e9T + 8.65e18T^{2} \)
17 \( 1 - 8.46e9T + 8.27e20T^{2} \)
19 \( 1 + 7.69e10T + 5.48e21T^{2} \)
23 \( 1 - 1.60e11T + 1.41e23T^{2} \)
31 \( 1 + 4.63e12T + 2.25e25T^{2} \)
37 \( 1 - 4.51e12T + 4.56e26T^{2} \)
41 \( 1 + 1.95e13T + 2.61e27T^{2} \)
43 \( 1 + 2.62e13T + 5.87e27T^{2} \)
47 \( 1 + 2.05e14T + 2.66e28T^{2} \)
53 \( 1 + 1.23e14T + 2.05e29T^{2} \)
59 \( 1 - 6.96e13T + 1.27e30T^{2} \)
61 \( 1 - 2.14e15T + 2.24e30T^{2} \)
67 \( 1 + 1.15e15T + 1.10e31T^{2} \)
71 \( 1 + 2.06e14T + 2.96e31T^{2} \)
73 \( 1 - 2.75e14T + 4.74e31T^{2} \)
79 \( 1 + 1.64e16T + 1.81e32T^{2} \)
83 \( 1 - 1.11e16T + 4.21e32T^{2} \)
89 \( 1 + 6.18e16T + 1.37e33T^{2} \)
97 \( 1 - 1.57e15T + 5.95e33T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.90248819359021077215526623752, −11.70777779325599863046714610406, −10.79091554868948151647872162740, −9.748779875686834646689898061312, −8.138900773760138402389225441399, −6.94781120120647277463505429781, −5.31628069589288092055024680706, −4.37396390791044862282198937403, −1.64312583781841867957991222632, −0.33166433993291021689051539078, 0.33166433993291021689051539078, 1.64312583781841867957991222632, 4.37396390791044862282198937403, 5.31628069589288092055024680706, 6.94781120120647277463505429781, 8.138900773760138402389225441399, 9.748779875686834646689898061312, 10.79091554868948151647872162740, 11.70777779325599863046714610406, 12.90248819359021077215526623752

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