Properties

Label 2-29-1.1-c17-0-9
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  − 553.·2-s − 346.·3-s + 1.75e5·4-s + 6.41e5·5-s + 1.91e5·6-s − 1.77e7·7-s − 2.43e7·8-s − 1.29e8·9-s − 3.55e8·10-s + 1.27e9·11-s − 6.06e7·12-s + 2.71e9·13-s + 9.79e9·14-s − 2.22e8·15-s − 9.46e9·16-s + 1.35e10·17-s + 7.13e10·18-s − 6.90e10·19-s + 1.12e11·20-s + 6.12e9·21-s − 7.05e11·22-s + 4.68e11·23-s + 8.43e9·24-s − 3.50e11·25-s − 1.50e12·26-s + 8.93e10·27-s − 3.10e12·28-s + ⋯
L(s)  = 1  − 1.52·2-s − 0.0304·3-s + 1.33·4-s + 0.734·5-s + 0.0465·6-s − 1.16·7-s − 0.513·8-s − 0.999·9-s − 1.12·10-s + 1.79·11-s − 0.0406·12-s + 0.922·13-s + 1.77·14-s − 0.0223·15-s − 0.551·16-s + 0.472·17-s + 1.52·18-s − 0.932·19-s + 0.981·20-s + 0.0353·21-s − 2.74·22-s + 1.24·23-s + 0.0156·24-s − 0.459·25-s − 1.41·26-s + 0.0608·27-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

\(L(9)\) \(\approx\) \(0.8597411167\)
\(L(\frac12)\) \(\approx\) \(0.8597411167\)
\(L(\frac{19}{2})\) 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 \( 1 + 553.T + 1.31e5T^{2} \)
3 \( 1 + 346.T + 1.29e8T^{2} \)
5 \( 1 - 6.41e5T + 7.62e11T^{2} \)
7 \( 1 + 1.77e7T + 2.32e14T^{2} \)
11 \( 1 - 1.27e9T + 5.05e17T^{2} \)
13 \( 1 - 2.71e9T + 8.65e18T^{2} \)
17 \( 1 - 1.35e10T + 8.27e20T^{2} \)
19 \( 1 + 6.90e10T + 5.48e21T^{2} \)
23 \( 1 - 4.68e11T + 1.41e23T^{2} \)
31 \( 1 + 7.33e12T + 2.25e25T^{2} \)
37 \( 1 + 2.41e13T + 4.56e26T^{2} \)
41 \( 1 + 3.62e13T + 2.61e27T^{2} \)
43 \( 1 - 1.17e14T + 5.87e27T^{2} \)
47 \( 1 + 2.50e14T + 2.66e28T^{2} \)
53 \( 1 - 8.36e14T + 2.05e29T^{2} \)
59 \( 1 - 6.11e14T + 1.27e30T^{2} \)
61 \( 1 - 1.58e15T + 2.24e30T^{2} \)
67 \( 1 + 1.78e15T + 1.10e31T^{2} \)
71 \( 1 + 4.49e15T + 2.96e31T^{2} \)
73 \( 1 + 1.35e16T + 4.74e31T^{2} \)
79 \( 1 - 1.86e16T + 1.81e32T^{2} \)
83 \( 1 - 1.42e16T + 4.21e32T^{2} \)
89 \( 1 - 6.42e16T + 1.37e33T^{2} \)
97 \( 1 - 5.31e16T + 5.95e33T^{2} \)
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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

−13.27265021236108007210353107580, −11.63382993548036061922641858405, −10.44191629746640287317701362809, −9.220610560113174944670899293655, −8.778476632149064319600718515355, −6.87747441803954297259827039578, −5.99783404568191868576898528720, −3.49021725401879573421041174589, −1.85103311738499055021103935940, −0.65550265170396099872496259602, 0.65550265170396099872496259602, 1.85103311738499055021103935940, 3.49021725401879573421041174589, 5.99783404568191868576898528720, 6.87747441803954297259827039578, 8.778476632149064319600718515355, 9.220610560113174944670899293655, 10.44191629746640287317701362809, 11.63382993548036061922641858405, 13.27265021236108007210353107580

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