Properties

Label 2-1-1.1-c63-0-0
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  + 2.54e9·2-s − 9.84e14·3-s − 2.75e18·4-s − 1.69e22·5-s − 2.50e24·6-s + 1.59e26·7-s − 3.04e28·8-s − 1.75e29·9-s − 4.30e31·10-s + 8.86e32·11-s + 2.70e33·12-s + 7.82e34·13-s + 4.05e35·14-s + 1.66e37·15-s − 5.21e37·16-s − 5.03e38·17-s − 4.45e38·18-s + 8.03e39·19-s + 4.65e40·20-s − 1.56e41·21-s + 2.25e42·22-s − 1.14e43·23-s + 2.99e43·24-s + 1.77e44·25-s + 1.99e44·26-s + 1.29e45·27-s − 4.38e44·28-s + ⋯
L(s)  = 1  + 0.837·2-s − 0.920·3-s − 0.298·4-s − 1.62·5-s − 0.770·6-s + 0.381·7-s − 1.08·8-s − 0.153·9-s − 1.36·10-s + 1.39·11-s + 0.274·12-s + 0.637·13-s + 0.319·14-s + 1.49·15-s − 0.612·16-s − 0.876·17-s − 0.128·18-s + 0.420·19-s + 0.484·20-s − 0.351·21-s + 1.16·22-s − 1.46·23-s + 1.00·24-s + 1.63·25-s + 0.533·26-s + 1.06·27-s − 0.113·28-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

\(L(32)\) \(\approx\) \(1.013537659\)
\(L(\frac12)\) \(\approx\) \(1.013537659\)
\(L(\frac{65}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.54e9T + 9.22e18T^{2} \)
3 \( 1 + 9.84e14T + 1.14e30T^{2} \)
5 \( 1 + 1.69e22T + 1.08e44T^{2} \)
7 \( 1 - 1.59e26T + 1.74e53T^{2} \)
11 \( 1 - 8.86e32T + 4.05e65T^{2} \)
13 \( 1 - 7.82e34T + 1.50e70T^{2} \)
17 \( 1 + 5.03e38T + 3.29e77T^{2} \)
19 \( 1 - 8.03e39T + 3.64e80T^{2} \)
23 \( 1 + 1.14e43T + 6.14e85T^{2} \)
29 \( 1 - 2.15e46T + 1.35e92T^{2} \)
31 \( 1 + 6.36e46T + 9.03e93T^{2} \)
37 \( 1 + 1.05e49T + 6.26e98T^{2} \)
41 \( 1 - 8.54e50T + 4.03e101T^{2} \)
43 \( 1 + 3.25e51T + 8.10e102T^{2} \)
47 \( 1 - 1.11e52T + 2.19e105T^{2} \)
53 \( 1 + 5.90e53T + 4.25e108T^{2} \)
59 \( 1 - 2.16e55T + 3.66e111T^{2} \)
61 \( 1 - 1.72e56T + 2.99e112T^{2} \)
67 \( 1 + 2.22e57T + 1.10e115T^{2} \)
71 \( 1 - 2.30e58T + 4.25e116T^{2} \)
73 \( 1 - 1.23e58T + 2.45e117T^{2} \)
79 \( 1 - 8.71e58T + 3.55e119T^{2} \)
83 \( 1 + 1.10e60T + 7.97e120T^{2} \)
89 \( 1 - 4.52e61T + 6.47e122T^{2} \)
97 \( 1 - 2.14e62T + 1.46e125T^{2} \)
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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

−17.84510587148979314760771788611, −15.94357929245443565257933538376, −14.31955506474695380971417929363, −12.13083194266712767185974727948, −11.40579155133014556535306382769, −8.521125339432963339889268140313, −6.36932168417662856012186571069, −4.64415932938404842791853805292, −3.65374290931023038947315699063, −0.60824199369184346752501954201, 0.60824199369184346752501954201, 3.65374290931023038947315699063, 4.64415932938404842791853805292, 6.36932168417662856012186571069, 8.521125339432963339889268140313, 11.40579155133014556535306382769, 12.13083194266712767185974727948, 14.31955506474695380971417929363, 15.94357929245443565257933538376, 17.84510587148979314760771788611

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