Properties

Label 2-29e2-1.1-c3-0-92
Degree $2$
Conductor $841$
Sign $1$
Analytic cond. $49.6206$
Root an. cond. $7.04418$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.414·2-s + 9.24·3-s − 7.82·4-s + 0.656·5-s − 3.82·6-s + 6.14·7-s + 6.55·8-s + 58.4·9-s − 0.272·10-s + 65.3·11-s − 72.3·12-s − 49.7·13-s − 2.54·14-s + 6.07·15-s + 59.9·16-s − 55.4·17-s − 24.2·18-s + 64.7·19-s − 5.14·20-s + 56.7·21-s − 27.0·22-s + 93.8·23-s + 60.5·24-s − 124.·25-s + 20.6·26-s + 290.·27-s − 48.0·28-s + ⋯
L(s)  = 1  − 0.146·2-s + 1.77·3-s − 0.978·4-s + 0.0587·5-s − 0.260·6-s + 0.331·7-s + 0.289·8-s + 2.16·9-s − 0.00860·10-s + 1.79·11-s − 1.74·12-s − 1.06·13-s − 0.0485·14-s + 0.104·15-s + 0.936·16-s − 0.791·17-s − 0.316·18-s + 0.781·19-s − 0.0574·20-s + 0.589·21-s − 0.262·22-s + 0.851·23-s + 0.515·24-s − 0.996·25-s + 0.155·26-s + 2.07·27-s − 0.324·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $1$
Analytic conductor: \(49.6206\)
Root analytic conductor: \(7.04418\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.525045624\)
\(L(\frac12)\) \(\approx\) \(3.525045624\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + 0.414T + 8T^{2} \)
3 \( 1 - 9.24T + 27T^{2} \)
5 \( 1 - 0.656T + 125T^{2} \)
7 \( 1 - 6.14T + 343T^{2} \)
11 \( 1 - 65.3T + 1.33e3T^{2} \)
13 \( 1 + 49.7T + 2.19e3T^{2} \)
17 \( 1 + 55.4T + 4.91e3T^{2} \)
19 \( 1 - 64.7T + 6.85e3T^{2} \)
23 \( 1 - 93.8T + 1.21e4T^{2} \)
31 \( 1 - 236.T + 2.97e4T^{2} \)
37 \( 1 + 76.8T + 5.06e4T^{2} \)
41 \( 1 + 215.T + 6.89e4T^{2} \)
43 \( 1 + 80.8T + 7.95e4T^{2} \)
47 \( 1 - 357.T + 1.03e5T^{2} \)
53 \( 1 - 328.T + 1.48e5T^{2} \)
59 \( 1 + 99.2T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 - 844.T + 3.00e5T^{2} \)
71 \( 1 + 378.T + 3.57e5T^{2} \)
73 \( 1 - 581.T + 3.89e5T^{2} \)
79 \( 1 - 353.T + 4.93e5T^{2} \)
83 \( 1 - 696.T + 5.71e5T^{2} \)
89 \( 1 + 1.11e3T + 7.04e5T^{2} \)
97 \( 1 - 805.T + 9.12e5T^{2} \)
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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

−9.589501026350119493705285476551, −8.941113540689246103215420912440, −8.388307113500313299338703885107, −7.50768133227673340277756820612, −6.68311863926631017039560372448, −5.01265273433794332958452392924, −4.16371286343589012715517510469, −3.44775024398054508960177529318, −2.19865151188190038502223309421, −1.06831714854413297563034374258, 1.06831714854413297563034374258, 2.19865151188190038502223309421, 3.44775024398054508960177529318, 4.16371286343589012715517510469, 5.01265273433794332958452392924, 6.68311863926631017039560372448, 7.50768133227673340277756820612, 8.388307113500313299338703885107, 8.941113540689246103215420912440, 9.589501026350119493705285476551

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