Properties

Label 2-1-1.1-c63-0-2
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  − 4.68e9·2-s − 3.45e14·3-s + 1.27e19·4-s − 3.68e21·5-s + 1.61e24·6-s + 6.76e26·7-s − 1.63e28·8-s − 1.02e30·9-s + 1.72e31·10-s − 4.60e32·11-s − 4.39e33·12-s − 1.39e35·13-s − 3.17e36·14-s + 1.27e36·15-s − 4.07e37·16-s + 6.98e38·17-s + 4.80e39·18-s − 2.26e40·19-s − 4.68e40·20-s − 2.34e41·21-s + 2.15e42·22-s + 9.79e42·23-s + 5.65e42·24-s − 9.48e43·25-s + 6.53e44·26-s + 7.50e44·27-s + 8.60e45·28-s + ⋯
L(s)  = 1  − 1.54·2-s − 0.323·3-s + 1.37·4-s − 0.353·5-s + 0.498·6-s + 1.62·7-s − 0.583·8-s − 0.895·9-s + 0.545·10-s − 0.723·11-s − 0.445·12-s − 1.13·13-s − 2.50·14-s + 0.114·15-s − 0.478·16-s + 1.21·17-s + 1.38·18-s − 1.18·19-s − 0.487·20-s − 0.524·21-s + 1.11·22-s + 1.24·23-s + 0.188·24-s − 0.874·25-s + 1.75·26-s + 0.612·27-s + 2.23·28-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

\(L(32)\) \(\approx\) \(0.6102351277\)
\(L(\frac12)\) \(\approx\) \(0.6102351277\)
\(L(\frac{65}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 4.68e9T + 9.22e18T^{2} \)
3 \( 1 + 3.45e14T + 1.14e30T^{2} \)
5 \( 1 + 3.68e21T + 1.08e44T^{2} \)
7 \( 1 - 6.76e26T + 1.74e53T^{2} \)
11 \( 1 + 4.60e32T + 4.05e65T^{2} \)
13 \( 1 + 1.39e35T + 1.50e70T^{2} \)
17 \( 1 - 6.98e38T + 3.29e77T^{2} \)
19 \( 1 + 2.26e40T + 3.64e80T^{2} \)
23 \( 1 - 9.79e42T + 6.14e85T^{2} \)
29 \( 1 - 1.02e46T + 1.35e92T^{2} \)
31 \( 1 + 7.88e46T + 9.03e93T^{2} \)
37 \( 1 + 4.47e48T + 6.26e98T^{2} \)
41 \( 1 + 2.61e50T + 4.03e101T^{2} \)
43 \( 1 - 1.36e51T + 8.10e102T^{2} \)
47 \( 1 + 1.83e52T + 2.19e105T^{2} \)
53 \( 1 - 3.07e54T + 4.25e108T^{2} \)
59 \( 1 - 6.07e55T + 3.66e111T^{2} \)
61 \( 1 - 2.59e56T + 2.99e112T^{2} \)
67 \( 1 + 2.81e57T + 1.10e115T^{2} \)
71 \( 1 + 2.35e58T + 4.25e116T^{2} \)
73 \( 1 - 3.66e58T + 2.45e117T^{2} \)
79 \( 1 - 9.89e59T + 3.55e119T^{2} \)
83 \( 1 - 3.72e59T + 7.97e120T^{2} \)
89 \( 1 - 2.65e61T + 6.47e122T^{2} \)
97 \( 1 - 2.06e62T + 1.46e125T^{2} \)
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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

−17.79506601430641543509648962802, −16.86783836802395123605928443560, −14.75113690352835432131230345710, −11.68596911820508741304314952304, −10.49602289767242460507648530691, −8.528091325463947058665729713447, −7.54164017761432726094566132719, −5.07839043795559135938970624058, −2.21468931928119782116171883813, −0.64534129392468597418821265851, 0.64534129392468597418821265851, 2.21468931928119782116171883813, 5.07839043795559135938970624058, 7.54164017761432726094566132719, 8.528091325463947058665729713447, 10.49602289767242460507648530691, 11.68596911820508741304314952304, 14.75113690352835432131230345710, 16.86783836802395123605928443560, 17.79506601430641543509648962802

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