Properties

Label 2-29-1.1-c17-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 290.·2-s − 1.63e4·3-s − 4.67e4·4-s + 8.05e5·5-s − 4.73e6·6-s + 2.28e7·7-s − 5.16e7·8-s + 1.36e8·9-s + 2.34e8·10-s − 1.07e9·11-s + 7.61e8·12-s − 1.42e9·13-s + 6.64e9·14-s − 1.31e10·15-s − 8.87e9·16-s − 7.26e9·17-s + 3.97e10·18-s + 2.69e10·19-s − 3.76e10·20-s − 3.73e11·21-s − 3.13e11·22-s − 1.74e11·23-s + 8.42e11·24-s − 1.13e11·25-s − 4.14e11·26-s − 1.25e11·27-s − 1.06e12·28-s + ⋯
L(s)  = 1  + 0.802·2-s − 1.43·3-s − 0.356·4-s + 0.922·5-s − 1.15·6-s + 1.49·7-s − 1.08·8-s + 1.05·9-s + 0.740·10-s − 1.51·11-s + 0.511·12-s − 0.485·13-s + 1.20·14-s − 1.32·15-s − 0.516·16-s − 0.252·17-s + 0.849·18-s + 0.363·19-s − 0.328·20-s − 2.15·21-s − 1.21·22-s − 0.465·23-s + 1.56·24-s − 0.149·25-s − 0.389·26-s − 0.0853·27-s − 0.534·28-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

\(L(9)\) \(\approx\) \(1.704654894\)
\(L(\frac12)\) \(\approx\) \(1.704654894\)
\(L(\frac{19}{2})\) 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 \( 1 - 290.T + 1.31e5T^{2} \)
3 \( 1 + 1.63e4T + 1.29e8T^{2} \)
5 \( 1 - 8.05e5T + 7.62e11T^{2} \)
7 \( 1 - 2.28e7T + 2.32e14T^{2} \)
11 \( 1 + 1.07e9T + 5.05e17T^{2} \)
13 \( 1 + 1.42e9T + 8.65e18T^{2} \)
17 \( 1 + 7.26e9T + 8.27e20T^{2} \)
19 \( 1 - 2.69e10T + 5.48e21T^{2} \)
23 \( 1 + 1.74e11T + 1.41e23T^{2} \)
31 \( 1 - 5.52e12T + 2.25e25T^{2} \)
37 \( 1 - 4.02e13T + 4.56e26T^{2} \)
41 \( 1 + 2.76e13T + 2.61e27T^{2} \)
43 \( 1 - 3.16e13T + 5.87e27T^{2} \)
47 \( 1 - 2.44e13T + 2.66e28T^{2} \)
53 \( 1 + 1.46e14T + 2.05e29T^{2} \)
59 \( 1 + 1.40e15T + 1.27e30T^{2} \)
61 \( 1 - 1.67e15T + 2.24e30T^{2} \)
67 \( 1 - 5.26e15T + 1.10e31T^{2} \)
71 \( 1 + 3.64e15T + 2.96e31T^{2} \)
73 \( 1 - 7.68e15T + 4.74e31T^{2} \)
79 \( 1 - 2.37e16T + 1.81e32T^{2} \)
83 \( 1 - 6.98e15T + 4.21e32T^{2} \)
89 \( 1 - 2.61e16T + 1.37e33T^{2} \)
97 \( 1 + 1.14e17T + 5.95e33T^{2} \)
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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

−13.26471636205459871909898911610, −12.11033178967171052215805404308, −11.06441379031678409788629077049, −9.885347836333025121324263084347, −8.012227602972860853360539459964, −6.09332777839673473983571708721, −5.21021155181699093748654261083, −4.66229688748345767553593239104, −2.34528073711613955706540793281, −0.68790028542265880243787767443, 0.68790028542265880243787767443, 2.34528073711613955706540793281, 4.66229688748345767553593239104, 5.21021155181699093748654261083, 6.09332777839673473983571708721, 8.012227602972860853360539459964, 9.885347836333025121324263084347, 11.06441379031678409788629077049, 12.11033178967171052215805404308, 13.26471636205459871909898911610

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