Properties

Label 2-15e2-1.1-c1-0-0
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.23·2-s + 3.00·4-s − 2.23·8-s − 0.999·16-s + 4.47·17-s + 4·19-s + 8.94·23-s + 8·31-s + 6.70·32-s − 10.0·34-s − 8.94·38-s − 20.0·46-s − 8.94·47-s − 7·49-s − 4.47·53-s + 2·61-s − 17.8·62-s − 13.0·64-s + 13.4·68-s + 12.0·76-s + 16·79-s − 17.8·83-s + 26.8·92-s + 20.0·94-s + 15.6·98-s + 10.0·106-s + 17.8·107-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.50·4-s − 0.790·8-s − 0.249·16-s + 1.08·17-s + 0.917·19-s + 1.86·23-s + 1.43·31-s + 1.18·32-s − 1.71·34-s − 1.45·38-s − 2.94·46-s − 1.30·47-s − 49-s − 0.614·53-s + 0.256·61-s − 2.27·62-s − 1.62·64-s + 1.62·68-s + 1.37·76-s + 1.80·79-s − 1.96·83-s + 2.79·92-s + 2.06·94-s + 1.58·98-s + 0.971·106-s + 1.72·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6097626324\)
\(L(\frac12)\) \(\approx\) \(0.6097626324\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 2T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 4.47T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8.94T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 + 4.47T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 + 17.8T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.84398909630952371043332939468, −11.05294965077240898604675348140, −10.04198369538576760343525987751, −9.397005801557129385093236827331, −8.361638436707407037404732372865, −7.54293240984768091665500147012, −6.55428878104586497175311860571, −5.01672023245190115764283599751, −3.02477398985994833328267227345, −1.17988259436947085351712516811, 1.17988259436947085351712516811, 3.02477398985994833328267227345, 5.01672023245190115764283599751, 6.55428878104586497175311860571, 7.54293240984768091665500147012, 8.361638436707407037404732372865, 9.397005801557129385093236827331, 10.04198369538576760343525987751, 11.05294965077240898604675348140, 11.84398909630952371043332939468

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