Properties

Degree $2$
Conductor $1$
Sign $-1$
Motivic weight $61$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.04e9·2-s + 1.78e14·3-s + 1.88e18·4-s − 2.27e20·5-s − 3.66e23·6-s − 4.42e25·7-s + 8.53e26·8-s − 9.51e28·9-s + 4.66e29·10-s + 7.45e31·11-s + 3.38e32·12-s + 1.57e34·13-s + 9.06e34·14-s − 4.08e34·15-s − 6.10e36·16-s − 3.16e37·17-s + 1.94e38·18-s + 4.06e38·19-s − 4.30e38·20-s − 7.92e39·21-s − 1.52e41·22-s + 5.44e40·23-s + 1.52e41·24-s − 4.28e42·25-s − 3.23e43·26-s − 3.97e43·27-s − 8.35e43·28-s + ⋯
L(s)  = 1  − 1.34·2-s + 0.501·3-s + 0.819·4-s − 0.109·5-s − 0.676·6-s − 0.742·7-s + 0.243·8-s − 0.748·9-s + 0.147·10-s + 1.28·11-s + 0.411·12-s + 1.67·13-s + 1.00·14-s − 0.0549·15-s − 1.14·16-s − 0.936·17-s + 1.00·18-s + 0.404·19-s − 0.0896·20-s − 0.372·21-s − 1.73·22-s + 0.159·23-s + 0.122·24-s − 0.988·25-s − 2.25·26-s − 0.877·27-s − 0.607·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(62-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+61/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Motivic weight: \(61\)
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :61/2),\ -1)\)

Particular Values

\(L(31)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{63}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.04e9T + 2.30e18T^{2} \)
3 \( 1 - 1.78e14T + 1.27e29T^{2} \)
5 \( 1 + 2.27e20T + 4.33e42T^{2} \)
7 \( 1 + 4.42e25T + 3.55e51T^{2} \)
11 \( 1 - 7.45e31T + 3.34e63T^{2} \)
13 \( 1 - 1.57e34T + 8.92e67T^{2} \)
17 \( 1 + 3.16e37T + 1.14e75T^{2} \)
19 \( 1 - 4.06e38T + 1.00e78T^{2} \)
23 \( 1 - 5.44e40T + 1.16e83T^{2} \)
29 \( 1 - 2.13e44T + 1.60e89T^{2} \)
31 \( 1 + 2.29e45T + 9.39e90T^{2} \)
37 \( 1 + 1.07e48T + 4.57e95T^{2} \)
41 \( 1 + 1.40e49T + 2.39e98T^{2} \)
43 \( 1 - 5.79e49T + 4.38e99T^{2} \)
47 \( 1 + 1.51e51T + 9.95e101T^{2} \)
53 \( 1 + 4.58e52T + 1.51e105T^{2} \)
59 \( 1 + 1.78e54T + 1.05e108T^{2} \)
61 \( 1 - 4.41e54T + 8.03e108T^{2} \)
67 \( 1 - 2.75e55T + 2.45e111T^{2} \)
71 \( 1 - 1.08e56T + 8.44e112T^{2} \)
73 \( 1 + 1.85e56T + 4.59e113T^{2} \)
79 \( 1 + 4.07e57T + 5.69e115T^{2} \)
83 \( 1 - 2.46e58T + 1.15e117T^{2} \)
89 \( 1 + 6.63e57T + 8.18e118T^{2} \)
97 \( 1 + 2.67e60T + 1.55e121T^{2} \)
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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

−17.52084445750528322790709380660, −15.98030677609940112156987834362, −13.74416012682528161665288772361, −11.22461445873856023968408214686, −9.360498338412850319100496424700, −8.435081671963490783505478108991, −6.52148749637659767175979247396, −3.57519995933766153820583028905, −1.56757984594175452865044515812, 0, 1.56757984594175452865044515812, 3.57519995933766153820583028905, 6.52148749637659767175979247396, 8.435081671963490783505478108991, 9.360498338412850319100496424700, 11.22461445873856023968408214686, 13.74416012682528161665288772361, 15.98030677609940112156987834362, 17.52084445750528322790709380660

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