Properties

Label 2-5-1.1-c21-0-3
Degree $2$
Conductor $5$
Sign $1$
Analytic cond. $13.9738$
Root an. cond. $3.73816$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.50e3·2-s − 1.81e5·3-s + 4.15e6·4-s + 9.76e6·5-s − 4.53e8·6-s + 9.51e8·7-s + 5.15e9·8-s + 2.24e10·9-s + 2.44e10·10-s + 6.26e10·11-s − 7.54e11·12-s + 3.65e11·13-s + 2.38e12·14-s − 1.77e12·15-s + 4.17e12·16-s − 4.06e12·17-s + 5.61e13·18-s − 3.33e12·19-s + 4.06e13·20-s − 1.72e14·21-s + 1.56e14·22-s + 1.41e14·23-s − 9.35e14·24-s + 9.53e13·25-s + 9.13e14·26-s − 2.17e15·27-s + 3.95e15·28-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.77·3-s + 1.98·4-s + 0.447·5-s − 3.06·6-s + 1.27·7-s + 1.69·8-s + 2.14·9-s + 0.772·10-s + 0.728·11-s − 3.51·12-s + 0.734·13-s + 2.19·14-s − 0.793·15-s + 0.949·16-s − 0.488·17-s + 3.70·18-s − 0.124·19-s + 0.886·20-s − 2.25·21-s + 1.25·22-s + 0.714·23-s − 3.01·24-s + 0.199·25-s + 1.26·26-s − 2.03·27-s + 2.52·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $1$
Analytic conductor: \(13.9738\)
Root analytic conductor: \(3.73816\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(3.602715408\)
\(L(\frac12)\) \(\approx\) \(3.602715408\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 9.76e6T \)
good2 \( 1 - 2.50e3T + 2.09e6T^{2} \)
3 \( 1 + 1.81e5T + 1.04e10T^{2} \)
7 \( 1 - 9.51e8T + 5.58e17T^{2} \)
11 \( 1 - 6.26e10T + 7.40e21T^{2} \)
13 \( 1 - 3.65e11T + 2.47e23T^{2} \)
17 \( 1 + 4.06e12T + 6.90e25T^{2} \)
19 \( 1 + 3.33e12T + 7.14e26T^{2} \)
23 \( 1 - 1.41e14T + 3.94e28T^{2} \)
29 \( 1 + 2.13e15T + 5.13e30T^{2} \)
31 \( 1 - 2.75e15T + 2.08e31T^{2} \)
37 \( 1 - 3.43e16T + 8.55e32T^{2} \)
41 \( 1 + 1.12e17T + 7.38e33T^{2} \)
43 \( 1 - 4.13e16T + 2.00e34T^{2} \)
47 \( 1 + 5.44e17T + 1.30e35T^{2} \)
53 \( 1 - 1.34e18T + 1.62e36T^{2} \)
59 \( 1 - 7.47e18T + 1.54e37T^{2} \)
61 \( 1 + 1.07e19T + 3.10e37T^{2} \)
67 \( 1 + 1.74e18T + 2.22e38T^{2} \)
71 \( 1 + 2.17e19T + 7.52e38T^{2} \)
73 \( 1 + 1.65e19T + 1.34e39T^{2} \)
79 \( 1 - 7.06e19T + 7.08e39T^{2} \)
83 \( 1 - 1.53e19T + 1.99e40T^{2} \)
89 \( 1 + 1.94e20T + 8.65e40T^{2} \)
97 \( 1 + 7.93e20T + 5.27e41T^{2} \)
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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

−17.94723742194866142304086696246, −16.63797142022275487980212749709, −14.97214485295283311601794706211, −13.27975010077361349831866451618, −11.76640738560790848346736173380, −11.00430035000056245379626896660, −6.62777943694871612797710607893, −5.45551714286228261970517123240, −4.37110406059634788583049541252, −1.48194548565018517145921995530, 1.48194548565018517145921995530, 4.37110406059634788583049541252, 5.45551714286228261970517123240, 6.62777943694871612797710607893, 11.00430035000056245379626896660, 11.76640738560790848346736173380, 13.27975010077361349831866451618, 14.97214485295283311601794706211, 16.63797142022275487980212749709, 17.94723742194866142304086696246

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