Properties

Label 2-29-1.1-c17-0-29
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  + 652.·2-s + 4.14e3·3-s + 2.95e5·4-s + 3.58e5·5-s + 2.70e6·6-s + 1.05e7·7-s + 1.07e8·8-s − 1.11e8·9-s + 2.33e8·10-s + 1.15e9·11-s + 1.22e9·12-s − 2.51e9·13-s + 6.86e9·14-s + 1.48e9·15-s + 3.12e10·16-s + 1.45e10·17-s − 7.31e10·18-s + 1.40e11·19-s + 1.05e11·20-s + 4.35e10·21-s + 7.55e11·22-s − 5.22e11·23-s + 4.43e11·24-s − 6.34e11·25-s − 1.64e12·26-s − 9.98e11·27-s + 3.10e12·28-s + ⋯
L(s)  = 1  + 1.80·2-s + 0.364·3-s + 2.25·4-s + 0.410·5-s + 0.657·6-s + 0.689·7-s + 2.25·8-s − 0.867·9-s + 0.739·10-s + 1.62·11-s + 0.820·12-s − 0.855·13-s + 1.24·14-s + 0.149·15-s + 1.81·16-s + 0.505·17-s − 1.56·18-s + 1.89·19-s + 0.923·20-s + 0.251·21-s + 2.93·22-s − 1.38·23-s + 0.822·24-s − 0.831·25-s − 1.54·26-s − 0.680·27-s + 1.55·28-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

\(L(9)\) \(\approx\) \(8.829044625\)
\(L(\frac12)\) \(\approx\) \(8.829044625\)
\(L(\frac{19}{2})\) 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 \( 1 - 652.T + 1.31e5T^{2} \)
3 \( 1 - 4.14e3T + 1.29e8T^{2} \)
5 \( 1 - 3.58e5T + 7.62e11T^{2} \)
7 \( 1 - 1.05e7T + 2.32e14T^{2} \)
11 \( 1 - 1.15e9T + 5.05e17T^{2} \)
13 \( 1 + 2.51e9T + 8.65e18T^{2} \)
17 \( 1 - 1.45e10T + 8.27e20T^{2} \)
19 \( 1 - 1.40e11T + 5.48e21T^{2} \)
23 \( 1 + 5.22e11T + 1.41e23T^{2} \)
31 \( 1 - 5.44e12T + 2.25e25T^{2} \)
37 \( 1 - 2.38e13T + 4.56e26T^{2} \)
41 \( 1 + 5.39e13T + 2.61e27T^{2} \)
43 \( 1 - 5.68e13T + 5.87e27T^{2} \)
47 \( 1 + 1.71e14T + 2.66e28T^{2} \)
53 \( 1 - 6.93e14T + 2.05e29T^{2} \)
59 \( 1 + 1.52e15T + 1.27e30T^{2} \)
61 \( 1 + 2.52e14T + 2.24e30T^{2} \)
67 \( 1 + 9.07e14T + 1.10e31T^{2} \)
71 \( 1 - 1.43e14T + 2.96e31T^{2} \)
73 \( 1 - 5.43e15T + 4.74e31T^{2} \)
79 \( 1 + 6.13e15T + 1.81e32T^{2} \)
83 \( 1 + 1.62e16T + 4.21e32T^{2} \)
89 \( 1 - 4.92e16T + 1.37e33T^{2} \)
97 \( 1 + 7.93e16T + 5.95e33T^{2} \)
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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

−13.83951308613720945278213511974, −11.93289676419376207477366288457, −11.67938219536357048020223748134, −9.666644977983306039066054646862, −7.77678020980747930210900299743, −6.27605545273809030954142438361, −5.25066688433097727535377404318, −3.96123332308021918544349544055, −2.77672449496463176858089199123, −1.52266995033413137981194983562, 1.52266995033413137981194983562, 2.77672449496463176858089199123, 3.96123332308021918544349544055, 5.25066688433097727535377404318, 6.27605545273809030954142438361, 7.77678020980747930210900299743, 9.666644977983306039066054646862, 11.67938219536357048020223748134, 11.93289676419376207477366288457, 13.83951308613720945278213511974

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