Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $105$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56e15·2-s + 7.24e24·3-s − 3.40e31·4-s − 1.17e36·5-s − 1.85e40·6-s − 1.82e44·7-s + 1.91e47·8-s − 7.27e49·9-s + 3.00e51·10-s + 6.21e54·11-s − 2.46e56·12-s + 9.87e57·13-s + 4.67e59·14-s − 8.48e60·15-s + 8.89e62·16-s + 1.10e64·17-s + 1.86e65·18-s + 1.61e67·19-s + 3.98e67·20-s − 1.32e69·21-s − 1.59e70·22-s − 2.39e71·23-s + 1.38e72·24-s − 2.32e73·25-s − 2.53e73·26-s − 1.43e75·27-s + 6.20e75·28-s + ⋯
L(s)  = 1  − 0.402·2-s + 0.647·3-s − 0.838·4-s − 0.235·5-s − 0.260·6-s − 0.783·7-s + 0.739·8-s − 0.581·9-s + 0.0948·10-s + 1.31·11-s − 0.542·12-s + 0.325·13-s + 0.315·14-s − 0.152·15-s + 0.540·16-s + 0.278·17-s + 0.233·18-s + 1.18·19-s + 0.197·20-s − 0.506·21-s − 0.530·22-s − 0.775·23-s + 0.478·24-s − 0.944·25-s − 0.130·26-s − 1.02·27-s + 0.656·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(105\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :105/2),\ -1)\)

Particular Values

\(L(53)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{107}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.56e15T + 4.05e31T^{2} \)
3 \( 1 - 7.24e24T + 1.25e50T^{2} \)
5 \( 1 + 1.17e36T + 2.46e73T^{2} \)
7 \( 1 + 1.82e44T + 5.43e88T^{2} \)
11 \( 1 - 6.21e54T + 2.21e109T^{2} \)
13 \( 1 - 9.87e57T + 9.20e116T^{2} \)
17 \( 1 - 1.10e64T + 1.57e129T^{2} \)
19 \( 1 - 1.61e67T + 1.85e134T^{2} \)
23 \( 1 + 2.39e71T + 9.58e142T^{2} \)
29 \( 1 - 4.95e76T + 3.56e153T^{2} \)
31 \( 1 - 8.41e76T + 3.91e156T^{2} \)
37 \( 1 - 2.33e82T + 4.58e164T^{2} \)
41 \( 1 - 3.56e84T + 2.19e169T^{2} \)
43 \( 1 + 2.88e85T + 3.26e171T^{2} \)
47 \( 1 - 8.77e86T + 3.71e175T^{2} \)
53 \( 1 + 1.32e90T + 1.11e181T^{2} \)
59 \( 1 - 9.30e92T + 8.69e185T^{2} \)
61 \( 1 + 9.36e93T + 2.88e187T^{2} \)
67 \( 1 + 4.39e95T + 5.46e191T^{2} \)
71 \( 1 + 2.93e97T + 2.41e194T^{2} \)
73 \( 1 + 6.72e97T + 4.45e195T^{2} \)
79 \( 1 + 7.95e99T + 1.78e199T^{2} \)
83 \( 1 - 6.25e100T + 3.18e201T^{2} \)
89 \( 1 - 3.11e102T + 4.85e204T^{2} \)
97 \( 1 + 3.96e104T + 4.08e208T^{2} \)
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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

−13.51901272130864475680765935145, −11.82226239066862034035907886284, −9.809514923245685914950115524611, −8.967593162309253346710299641169, −7.76250669663305613210077564119, −5.98083503443993720406932585882, −4.13367633193593308154224536720, −3.13596988314855035476847925613, −1.28103608680477674336644983983, 0, 1.28103608680477674336644983983, 3.13596988314855035476847925613, 4.13367633193593308154224536720, 5.98083503443993720406932585882, 7.76250669663305613210077564119, 8.967593162309253346710299641169, 9.809514923245685914950115524611, 11.82226239066862034035907886284, 13.51901272130864475680765935145

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