Properties

Label 2-45-1.1-c21-0-7
Degree $2$
Conductor $45$
Sign $1$
Analytic cond. $125.764$
Root an. cond. $11.2144$
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  − 625.·2-s − 1.70e6·4-s − 9.76e6·5-s + 3.78e8·7-s + 2.37e9·8-s + 6.10e9·10-s + 3.73e10·11-s − 9.27e11·13-s − 2.36e11·14-s + 2.09e12·16-s + 1.40e13·17-s + 3.31e13·19-s + 1.66e13·20-s − 2.33e13·22-s − 3.30e14·23-s + 9.53e13·25-s + 5.79e14·26-s − 6.46e14·28-s − 2.35e15·29-s + 3.97e15·31-s − 6.29e15·32-s − 8.77e15·34-s − 3.69e15·35-s − 4.11e16·37-s − 2.07e16·38-s − 2.32e16·40-s + 5.23e16·41-s + ⋯
L(s)  = 1  − 0.431·2-s − 0.813·4-s − 0.447·5-s + 0.506·7-s + 0.783·8-s + 0.193·10-s + 0.433·11-s − 1.86·13-s − 0.218·14-s + 0.475·16-s + 1.68·17-s + 1.23·19-s + 0.363·20-s − 0.187·22-s − 1.66·23-s + 0.199·25-s + 0.805·26-s − 0.412·28-s − 1.04·29-s + 0.870·31-s − 0.988·32-s − 0.729·34-s − 0.226·35-s − 1.40·37-s − 0.534·38-s − 0.350·40-s + 0.608·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(125.764\)
Root analytic conductor: \(11.2144\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.9820836994\)
\(L(\frac12)\) \(\approx\) \(0.9820836994\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + 9.76e6T \)
good2 \( 1 + 625.T + 2.09e6T^{2} \)
7 \( 1 - 3.78e8T + 5.58e17T^{2} \)
11 \( 1 - 3.73e10T + 7.40e21T^{2} \)
13 \( 1 + 9.27e11T + 2.47e23T^{2} \)
17 \( 1 - 1.40e13T + 6.90e25T^{2} \)
19 \( 1 - 3.31e13T + 7.14e26T^{2} \)
23 \( 1 + 3.30e14T + 3.94e28T^{2} \)
29 \( 1 + 2.35e15T + 5.13e30T^{2} \)
31 \( 1 - 3.97e15T + 2.08e31T^{2} \)
37 \( 1 + 4.11e16T + 8.55e32T^{2} \)
41 \( 1 - 5.23e16T + 7.38e33T^{2} \)
43 \( 1 - 2.10e17T + 2.00e34T^{2} \)
47 \( 1 + 5.77e17T + 1.30e35T^{2} \)
53 \( 1 + 1.16e18T + 1.62e36T^{2} \)
59 \( 1 + 4.17e18T + 1.54e37T^{2} \)
61 \( 1 - 9.77e18T + 3.10e37T^{2} \)
67 \( 1 + 1.53e19T + 2.22e38T^{2} \)
71 \( 1 - 6.75e18T + 7.52e38T^{2} \)
73 \( 1 + 1.58e19T + 1.34e39T^{2} \)
79 \( 1 + 1.48e19T + 7.08e39T^{2} \)
83 \( 1 - 6.30e19T + 1.99e40T^{2} \)
89 \( 1 + 1.22e20T + 8.65e40T^{2} \)
97 \( 1 - 5.41e20T + 5.27e41T^{2} \)
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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

−11.74281092472578239140261441204, −10.09686928276854919567939588192, −9.483459945844660153219122945998, −7.991392306224711397445918015734, −7.47400900377655106709746858276, −5.49282207059025500609540872302, −4.55586248848710531257708936492, −3.34414659808914850725861468710, −1.67414911027982636189640623942, −0.50659759460686242462831000900, 0.50659759460686242462831000900, 1.67414911027982636189640623942, 3.34414659808914850725861468710, 4.55586248848710531257708936492, 5.49282207059025500609540872302, 7.47400900377655106709746858276, 7.991392306224711397445918015734, 9.483459945844660153219122945998, 10.09686928276854919567939588192, 11.74281092472578239140261441204

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