Properties

Label 2-1-1.1-c63-0-1
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $25.1360$
Root an. cond. $5.01359$
Motivic weight $63$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9.25e8·2-s − 5.88e14·3-s − 8.36e18·4-s + 1.60e22·5-s + 5.44e23·6-s − 7.44e26·7-s + 1.62e28·8-s − 7.98e29·9-s − 1.48e31·10-s + 1.34e32·11-s + 4.91e33·12-s − 1.63e35·13-s + 6.89e35·14-s − 9.42e36·15-s + 6.20e37·16-s − 2.10e38·17-s + 7.39e38·18-s + 1.37e40·19-s − 1.34e41·20-s + 4.37e41·21-s − 1.24e41·22-s + 6.54e42·23-s − 9.57e42·24-s + 1.48e44·25-s + 1.50e44·26-s + 1.14e45·27-s + 6.22e45·28-s + ⋯
L(s)  = 1  − 0.304·2-s − 0.549·3-s − 0.907·4-s + 1.53·5-s + 0.167·6-s − 1.78·7-s + 0.581·8-s − 0.697·9-s − 0.469·10-s + 0.210·11-s + 0.498·12-s − 1.32·13-s + 0.543·14-s − 0.845·15-s + 0.729·16-s − 0.366·17-s + 0.212·18-s + 0.719·19-s − 1.39·20-s + 0.980·21-s − 0.0641·22-s + 0.835·23-s − 0.319·24-s + 1.36·25-s + 0.404·26-s + 0.933·27-s + 1.61·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(25.1360\)
Root analytic conductor: \(5.01359\)
Motivic weight: \(63\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :63/2),\ 1)\)

Particular Values

\(L(32)\) \(\approx\) \(0.8257887907\)
\(L(\frac12)\) \(\approx\) \(0.8257887907\)
\(L(\frac{65}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 9.25e8T + 9.22e18T^{2} \)
3 \( 1 + 5.88e14T + 1.14e30T^{2} \)
5 \( 1 - 1.60e22T + 1.08e44T^{2} \)
7 \( 1 + 7.44e26T + 1.74e53T^{2} \)
11 \( 1 - 1.34e32T + 4.05e65T^{2} \)
13 \( 1 + 1.63e35T + 1.50e70T^{2} \)
17 \( 1 + 2.10e38T + 3.29e77T^{2} \)
19 \( 1 - 1.37e40T + 3.64e80T^{2} \)
23 \( 1 - 6.54e42T + 6.14e85T^{2} \)
29 \( 1 - 8.71e45T + 1.35e92T^{2} \)
31 \( 1 - 7.38e46T + 9.03e93T^{2} \)
37 \( 1 - 3.54e48T + 6.26e98T^{2} \)
41 \( 1 - 3.75e50T + 4.03e101T^{2} \)
43 \( 1 - 1.72e50T + 8.10e102T^{2} \)
47 \( 1 + 2.40e52T + 2.19e105T^{2} \)
53 \( 1 + 2.64e54T + 4.25e108T^{2} \)
59 \( 1 - 7.37e55T + 3.66e111T^{2} \)
61 \( 1 + 3.17e55T + 2.99e112T^{2} \)
67 \( 1 - 1.19e57T + 1.10e115T^{2} \)
71 \( 1 - 1.32e58T + 4.25e116T^{2} \)
73 \( 1 - 3.25e58T + 2.45e117T^{2} \)
79 \( 1 - 7.17e58T + 3.55e119T^{2} \)
83 \( 1 - 3.84e60T + 7.97e120T^{2} \)
89 \( 1 + 3.50e61T + 6.47e122T^{2} \)
97 \( 1 + 5.88e62T + 1.46e125T^{2} \)
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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

−17.68424381185167711497329760694, −16.76696107115501479762168938573, −13.97831662866570681932682459532, −12.72798324229271226363525222543, −10.02979235712034670566762929387, −9.267756777022281022914557827508, −6.47207300135663117431492117326, −5.19326363783780132194148380768, −2.80685379906930605441069033119, −0.62694026687984783057215458063, 0.62694026687984783057215458063, 2.80685379906930605441069033119, 5.19326363783780132194148380768, 6.47207300135663117431492117326, 9.267756777022281022914557827508, 10.02979235712034670566762929387, 12.72798324229271226363525222543, 13.97831662866570681932682459532, 16.76696107115501479762168938573, 17.68424381185167711497329760694

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