Properties

Label 2-240-1.1-c9-0-4
Degree $2$
Conductor $240$
Sign $1$
Analytic cond. $123.608$
Root an. cond. $11.1179$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 81·3-s − 625·5-s + 4.01e3·7-s + 6.56e3·9-s − 8.48e4·11-s + 1.19e5·13-s + 5.06e4·15-s + 1.16e5·17-s + 2.34e5·19-s − 3.24e5·21-s − 2.34e6·23-s + 3.90e5·25-s − 5.31e5·27-s − 4.64e5·29-s + 5.11e6·31-s + 6.87e6·33-s − 2.50e6·35-s + 8.69e6·37-s − 9.67e6·39-s − 9.05e6·41-s − 8.63e6·43-s − 4.10e6·45-s − 3.31e7·47-s − 2.42e7·49-s − 9.47e6·51-s − 6.41e7·53-s + 5.30e7·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 0.631·7-s + 0.333·9-s − 1.74·11-s + 1.15·13-s + 0.258·15-s + 0.339·17-s + 0.413·19-s − 0.364·21-s − 1.74·23-s + 0.200·25-s − 0.192·27-s − 0.121·29-s + 0.995·31-s + 1.00·33-s − 0.282·35-s + 0.762·37-s − 0.669·39-s − 0.500·41-s − 0.385·43-s − 0.149·45-s − 0.990·47-s − 0.601·49-s − 0.196·51-s − 1.11·53-s + 0.781·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(123.608\)
Root analytic conductor: \(11.1179\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.250067177\)
\(L(\frac12)\) \(\approx\) \(1.250067177\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 81T \)
5 \( 1 + 625T \)
good7 \( 1 - 4.01e3T + 4.03e7T^{2} \)
11 \( 1 + 8.48e4T + 2.35e9T^{2} \)
13 \( 1 - 1.19e5T + 1.06e10T^{2} \)
17 \( 1 - 1.16e5T + 1.18e11T^{2} \)
19 \( 1 - 2.34e5T + 3.22e11T^{2} \)
23 \( 1 + 2.34e6T + 1.80e12T^{2} \)
29 \( 1 + 4.64e5T + 1.45e13T^{2} \)
31 \( 1 - 5.11e6T + 2.64e13T^{2} \)
37 \( 1 - 8.69e6T + 1.29e14T^{2} \)
41 \( 1 + 9.05e6T + 3.27e14T^{2} \)
43 \( 1 + 8.63e6T + 5.02e14T^{2} \)
47 \( 1 + 3.31e7T + 1.11e15T^{2} \)
53 \( 1 + 6.41e7T + 3.29e15T^{2} \)
59 \( 1 + 1.49e8T + 8.66e15T^{2} \)
61 \( 1 - 1.54e8T + 1.16e16T^{2} \)
67 \( 1 + 2.72e8T + 2.72e16T^{2} \)
71 \( 1 - 3.56e8T + 4.58e16T^{2} \)
73 \( 1 - 2.06e8T + 5.88e16T^{2} \)
79 \( 1 - 4.04e8T + 1.19e17T^{2} \)
83 \( 1 - 5.17e6T + 1.86e17T^{2} \)
89 \( 1 - 4.32e8T + 3.50e17T^{2} \)
97 \( 1 + 1.32e9T + 7.60e17T^{2} \)
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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

−10.66683304307045214018557953336, −9.779814413332066795473311667615, −8.104019346006165527132416540627, −7.929227391751733411359698196316, −6.39788060244603872973061895633, −5.42143717186757852901116049022, −4.50286353104763628960535066187, −3.23076233337456084916915390970, −1.80887010508796401555275439371, −0.53087056877413037517227093137, 0.53087056877413037517227093137, 1.80887010508796401555275439371, 3.23076233337456084916915390970, 4.50286353104763628960535066187, 5.42143717186757852901116049022, 6.39788060244603872973061895633, 7.929227391751733411359698196316, 8.104019346006165527132416540627, 9.779814413332066795473311667615, 10.66683304307045214018557953336

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