Properties

Label 2-45-1.1-c21-0-0
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  − 1.48e3·2-s + 1.16e5·4-s − 9.76e6·5-s − 1.26e9·7-s + 2.94e9·8-s + 1.45e10·10-s − 4.25e10·11-s + 7.14e11·13-s + 1.88e12·14-s − 4.62e12·16-s − 9.08e12·17-s − 2.62e13·19-s − 1.14e12·20-s + 6.33e13·22-s + 2.96e13·23-s + 9.53e13·25-s − 1.06e15·26-s − 1.48e14·28-s − 4.45e15·29-s + 4.93e14·31-s + 7.09e14·32-s + 1.35e16·34-s + 1.23e16·35-s + 8.82e15·37-s + 3.90e16·38-s − 2.87e16·40-s − 1.17e17·41-s + ⋯
L(s)  = 1  − 1.02·2-s + 0.0557·4-s − 0.447·5-s − 1.69·7-s + 0.970·8-s + 0.459·10-s − 0.495·11-s + 1.43·13-s + 1.74·14-s − 1.05·16-s − 1.09·17-s − 0.983·19-s − 0.0249·20-s + 0.508·22-s + 0.149·23-s + 0.199·25-s − 1.47·26-s − 0.0946·28-s − 1.96·29-s + 0.108·31-s + 0.111·32-s + 1.12·34-s + 0.758·35-s + 0.301·37-s + 1.01·38-s − 0.433·40-s − 1.36·41-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

\(L(11)\) \(\approx\) \(0.003370100382\)
\(L(\frac12)\) \(\approx\) \(0.003370100382\)
\(L(\frac{23}{2})\) 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 \( 1 + 1.48e3T + 2.09e6T^{2} \)
7 \( 1 + 1.26e9T + 5.58e17T^{2} \)
11 \( 1 + 4.25e10T + 7.40e21T^{2} \)
13 \( 1 - 7.14e11T + 2.47e23T^{2} \)
17 \( 1 + 9.08e12T + 6.90e25T^{2} \)
19 \( 1 + 2.62e13T + 7.14e26T^{2} \)
23 \( 1 - 2.96e13T + 3.94e28T^{2} \)
29 \( 1 + 4.45e15T + 5.13e30T^{2} \)
31 \( 1 - 4.93e14T + 2.08e31T^{2} \)
37 \( 1 - 8.82e15T + 8.55e32T^{2} \)
41 \( 1 + 1.17e17T + 7.38e33T^{2} \)
43 \( 1 + 4.77e16T + 2.00e34T^{2} \)
47 \( 1 + 3.81e17T + 1.30e35T^{2} \)
53 \( 1 + 1.01e18T + 1.62e36T^{2} \)
59 \( 1 - 8.26e17T + 1.54e37T^{2} \)
61 \( 1 + 6.60e18T + 3.10e37T^{2} \)
67 \( 1 + 2.57e19T + 2.22e38T^{2} \)
71 \( 1 + 3.32e19T + 7.52e38T^{2} \)
73 \( 1 + 2.66e19T + 1.34e39T^{2} \)
79 \( 1 + 7.85e19T + 7.08e39T^{2} \)
83 \( 1 + 1.68e20T + 1.99e40T^{2} \)
89 \( 1 + 6.58e19T + 8.65e40T^{2} \)
97 \( 1 + 1.13e21T + 5.27e41T^{2} \)
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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

−11.20276644605456914388909685622, −10.29779798429274518786188744013, −9.200081323425872472161285737933, −8.453199949195339498434383529489, −7.11931413950061897356061873488, −6.08379063346739706529498170478, −4.28034506390677916071463978657, −3.21115415125424645186078196694, −1.63762515646515751208863686285, −0.03115564611230382885270971954, 0.03115564611230382885270971954, 1.63762515646515751208863686285, 3.21115415125424645186078196694, 4.28034506390677916071463978657, 6.08379063346739706529498170478, 7.11931413950061897356061873488, 8.453199949195339498434383529489, 9.200081323425872472161285737933, 10.29779798429274518786188744013, 11.20276644605456914388909685622

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