Properties

Label 2-15e2-1.1-c11-0-39
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 83.7·2-s + 4.96e3·4-s − 1.59e4·7-s − 2.44e5·8-s − 3.39e5·11-s − 2.02e6·13-s + 1.33e6·14-s + 1.02e7·16-s − 2.45e6·17-s − 4.08e6·19-s + 2.84e7·22-s + 2.86e7·23-s + 1.69e8·26-s − 7.92e7·28-s + 9.41e6·29-s + 2.99e8·31-s − 3.60e8·32-s + 2.05e8·34-s + 4.57e8·37-s + 3.42e8·38-s − 1.83e8·41-s − 6.56e8·43-s − 1.68e9·44-s − 2.39e9·46-s − 1.97e8·47-s − 1.72e9·49-s − 1.00e10·52-s + ⋯
L(s)  = 1  − 1.85·2-s + 2.42·4-s − 0.359·7-s − 2.63·8-s − 0.636·11-s − 1.51·13-s + 0.664·14-s + 2.44·16-s − 0.418·17-s − 0.378·19-s + 1.17·22-s + 0.928·23-s + 2.79·26-s − 0.870·28-s + 0.0852·29-s + 1.87·31-s − 1.89·32-s + 0.774·34-s + 1.08·37-s + 0.700·38-s − 0.247·41-s − 0.681·43-s − 1.54·44-s − 1.71·46-s − 0.125·47-s − 0.870·49-s − 3.66·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(172.877\)
Root analytic conductor: \(13.1482\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 \)
good2 \( 1 + 83.7T + 2.04e3T^{2} \)
7 \( 1 + 1.59e4T + 1.97e9T^{2} \)
11 \( 1 + 3.39e5T + 2.85e11T^{2} \)
13 \( 1 + 2.02e6T + 1.79e12T^{2} \)
17 \( 1 + 2.45e6T + 3.42e13T^{2} \)
19 \( 1 + 4.08e6T + 1.16e14T^{2} \)
23 \( 1 - 2.86e7T + 9.52e14T^{2} \)
29 \( 1 - 9.41e6T + 1.22e16T^{2} \)
31 \( 1 - 2.99e8T + 2.54e16T^{2} \)
37 \( 1 - 4.57e8T + 1.77e17T^{2} \)
41 \( 1 + 1.83e8T + 5.50e17T^{2} \)
43 \( 1 + 6.56e8T + 9.29e17T^{2} \)
47 \( 1 + 1.97e8T + 2.47e18T^{2} \)
53 \( 1 - 5.15e9T + 9.26e18T^{2} \)
59 \( 1 - 6.62e8T + 3.01e19T^{2} \)
61 \( 1 - 5.58e8T + 4.35e19T^{2} \)
67 \( 1 + 1.01e10T + 1.22e20T^{2} \)
71 \( 1 + 1.78e10T + 2.31e20T^{2} \)
73 \( 1 - 2.33e10T + 3.13e20T^{2} \)
79 \( 1 - 1.24e10T + 7.47e20T^{2} \)
83 \( 1 - 3.37e10T + 1.28e21T^{2} \)
89 \( 1 + 2.94e10T + 2.77e21T^{2} \)
97 \( 1 - 1.13e11T + 7.15e21T^{2} \)
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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

−9.832077191862921702336580199818, −8.882838930976486525667724732387, −7.981144482154789933688345489069, −7.15686438945354711273778416985, −6.31439808697437873499341140938, −4.83043184974620726268123541900, −2.89295716250569474656555462666, −2.21210263452559806785023103956, −0.857342188108510598842367918745, 0, 0.857342188108510598842367918745, 2.21210263452559806785023103956, 2.89295716250569474656555462666, 4.83043184974620726268123541900, 6.31439808697437873499341140938, 7.15686438945354711273778416985, 7.981144482154789933688345489069, 8.882838930976486525667724732387, 9.832077191862921702336580199818

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