Properties

Label 2-1-1.1-c27-0-1
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $4.61855$
Root an. cond. $2.14908$
Motivic weight $27$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.04e4·2-s + 2.15e6·3-s − 2.53e7·4-s + 4.86e9·5-s + 2.24e10·6-s + 2.14e10·7-s − 1.66e12·8-s − 2.98e12·9-s + 5.07e13·10-s − 6.85e13·11-s − 5.46e13·12-s − 5.86e14·13-s + 2.24e14·14-s + 1.04e16·15-s − 1.39e16·16-s − 2.42e16·17-s − 3.11e16·18-s + 2.43e17·19-s − 1.23e17·20-s + 4.62e16·21-s − 7.15e17·22-s + 1.25e17·23-s − 3.58e18·24-s + 1.62e19·25-s − 6.12e18·26-s − 2.28e19·27-s − 5.44e17·28-s + ⋯
L(s)  = 1  + 0.900·2-s + 0.780·3-s − 0.188·4-s + 1.78·5-s + 0.702·6-s + 0.0837·7-s − 1.07·8-s − 0.391·9-s + 1.60·10-s − 0.598·11-s − 0.147·12-s − 0.537·13-s + 0.0754·14-s + 1.39·15-s − 0.775·16-s − 0.594·17-s − 0.352·18-s + 1.32·19-s − 0.336·20-s + 0.0653·21-s − 0.539·22-s + 0.0519·23-s − 0.835·24-s + 2.17·25-s − 0.483·26-s − 1.08·27-s − 0.0158·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(28-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+27/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(4.61855\)
Root analytic conductor: \(2.14908\)
Motivic weight: \(27\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(2.872749561\)
\(L(\frac12)\) \(\approx\) \(2.872749561\)
\(L(\frac{29}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 1.04e4T + 1.34e8T^{2} \)
3 \( 1 - 2.15e6T + 7.62e12T^{2} \)
5 \( 1 - 4.86e9T + 7.45e18T^{2} \)
7 \( 1 - 2.14e10T + 6.57e22T^{2} \)
11 \( 1 + 6.85e13T + 1.31e28T^{2} \)
13 \( 1 + 5.86e14T + 1.19e30T^{2} \)
17 \( 1 + 2.42e16T + 1.66e33T^{2} \)
19 \( 1 - 2.43e17T + 3.36e34T^{2} \)
23 \( 1 - 1.25e17T + 5.84e36T^{2} \)
29 \( 1 + 1.25e19T + 3.05e39T^{2} \)
31 \( 1 + 6.23e19T + 1.84e40T^{2} \)
37 \( 1 - 3.67e20T + 2.19e42T^{2} \)
41 \( 1 - 3.60e21T + 3.50e43T^{2} \)
43 \( 1 - 1.19e22T + 1.26e44T^{2} \)
47 \( 1 - 8.26e21T + 1.40e45T^{2} \)
53 \( 1 - 1.60e23T + 3.59e46T^{2} \)
59 \( 1 - 1.09e24T + 6.50e47T^{2} \)
61 \( 1 + 1.32e24T + 1.59e48T^{2} \)
67 \( 1 - 6.27e24T + 2.01e49T^{2} \)
71 \( 1 + 1.65e25T + 9.63e49T^{2} \)
73 \( 1 + 7.93e24T + 2.04e50T^{2} \)
79 \( 1 - 1.61e25T + 1.72e51T^{2} \)
83 \( 1 - 1.00e26T + 6.53e51T^{2} \)
89 \( 1 + 1.72e26T + 4.30e52T^{2} \)
97 \( 1 + 1.03e27T + 4.39e53T^{2} \)
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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

−26.24073429598295890148619961947, −24.64414644958356609087659196152, −22.21892371120360750818924346610, −20.79737968853638185831337102591, −17.84624530116015061782129782870, −14.39270451750690737940576636597, −13.30841394829815483779194433379, −9.348954137281175526758391198685, −5.49504866683922240527961455874, −2.60840721330549053446694296621, 2.60840721330549053446694296621, 5.49504866683922240527961455874, 9.348954137281175526758391198685, 13.30841394829815483779194433379, 14.39270451750690737940576636597, 17.84624530116015061782129782870, 20.79737968853638185831337102591, 22.21892371120360750818924346610, 24.64414644958356609087659196152, 26.24073429598295890148619961947

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