Properties

Label 2-29-1.1-c11-0-21
Degree $2$
Conductor $29$
Sign $-1$
Analytic cond. $22.2819$
Root an. cond. $4.72037$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 62.2·2-s − 384.·3-s + 1.83e3·4-s + 1.37e3·5-s − 2.39e4·6-s + 3.30e4·7-s − 1.35e4·8-s − 2.91e4·9-s + 8.57e4·10-s − 8.00e5·11-s − 7.03e5·12-s + 1.48e6·13-s + 2.05e6·14-s − 5.29e5·15-s − 4.59e6·16-s − 1.13e7·17-s − 1.81e6·18-s − 9.78e6·19-s + 2.52e6·20-s − 1.27e7·21-s − 4.98e7·22-s + 5.46e5·23-s + 5.22e6·24-s − 4.69e7·25-s + 9.25e7·26-s + 7.93e7·27-s + 6.04e7·28-s + ⋯
L(s)  = 1  + 1.37·2-s − 0.913·3-s + 0.893·4-s + 0.197·5-s − 1.25·6-s + 0.743·7-s − 0.146·8-s − 0.164·9-s + 0.271·10-s − 1.49·11-s − 0.816·12-s + 1.10·13-s + 1.02·14-s − 0.180·15-s − 1.09·16-s − 1.93·17-s − 0.226·18-s − 0.906·19-s + 0.176·20-s − 0.679·21-s − 2.06·22-s + 0.0176·23-s + 0.133·24-s − 0.961·25-s + 1.52·26-s + 1.06·27-s + 0.664·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-1$
Analytic conductor: \(22.2819\)
Root analytic conductor: \(4.72037\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 - 2.05e7T \)
good2 \( 1 - 62.2T + 2.04e3T^{2} \)
3 \( 1 + 384.T + 1.77e5T^{2} \)
5 \( 1 - 1.37e3T + 4.88e7T^{2} \)
7 \( 1 - 3.30e4T + 1.97e9T^{2} \)
11 \( 1 + 8.00e5T + 2.85e11T^{2} \)
13 \( 1 - 1.48e6T + 1.79e12T^{2} \)
17 \( 1 + 1.13e7T + 3.42e13T^{2} \)
19 \( 1 + 9.78e6T + 1.16e14T^{2} \)
23 \( 1 - 5.46e5T + 9.52e14T^{2} \)
31 \( 1 - 4.17e6T + 2.54e16T^{2} \)
37 \( 1 - 3.50e8T + 1.77e17T^{2} \)
41 \( 1 - 7.34e8T + 5.50e17T^{2} \)
43 \( 1 + 1.74e9T + 9.29e17T^{2} \)
47 \( 1 - 1.70e9T + 2.47e18T^{2} \)
53 \( 1 - 2.30e9T + 9.26e18T^{2} \)
59 \( 1 + 2.84e8T + 3.01e19T^{2} \)
61 \( 1 - 5.13e9T + 4.35e19T^{2} \)
67 \( 1 + 3.85e9T + 1.22e20T^{2} \)
71 \( 1 - 2.82e10T + 2.31e20T^{2} \)
73 \( 1 + 1.81e10T + 3.13e20T^{2} \)
79 \( 1 + 3.94e10T + 7.47e20T^{2} \)
83 \( 1 - 1.27e10T + 1.28e21T^{2} \)
89 \( 1 - 4.46e10T + 2.77e21T^{2} \)
97 \( 1 + 7.40e10T + 7.15e21T^{2} \)
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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

−13.66503578164396657718120411603, −12.94007247475097494298616766339, −11.48245844543128752891157656516, −10.80941303176049225843541504190, −8.486070669408914160620511493035, −6.40530109529597682147850237669, −5.41267843264848241503571637047, −4.33908871653396548875867134884, −2.38301702766475081334257295754, 0, 2.38301702766475081334257295754, 4.33908871653396548875867134884, 5.41267843264848241503571637047, 6.40530109529597682147850237669, 8.486070669408914160620511493035, 10.80941303176049225843541504190, 11.48245844543128752891157656516, 12.94007247475097494298616766339, 13.66503578164396657718120411603

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