Properties

Label 2-15e2-1.1-c1-0-1
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 3·8-s + 4·11-s + 2·13-s − 16-s + 2·17-s + 4·19-s − 4·22-s − 2·26-s + 2·29-s − 5·32-s − 2·34-s + 10·37-s − 4·38-s − 10·41-s − 4·43-s − 4·44-s + 8·47-s − 7·49-s − 2·52-s − 10·53-s − 2·58-s + 4·59-s − 2·61-s + 7·64-s − 12·67-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1.20·11-s + 0.554·13-s − 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s − 0.392·26-s + 0.371·29-s − 0.883·32-s − 0.342·34-s + 1.64·37-s − 0.648·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 49-s − 0.277·52-s − 1.37·53-s − 0.262·58-s + 0.520·59-s − 0.256·61-s + 7/8·64-s − 1.46·67-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: yes
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8242959390\)
\(L(\frac12)\) \(\approx\) \(0.8242959390\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.10713864521281605598821868486, −11.19897971912574422599178738873, −10.04517050900761317536685250745, −9.327820441539330420804608673051, −8.455181921164028207665051683361, −7.46261028773657489832215471574, −6.20681526904542088835730053039, −4.79492945983632365647604814265, −3.55714635244582915245852508954, −1.27025970316177507740293768624, 1.27025970316177507740293768624, 3.55714635244582915245852508954, 4.79492945983632365647604814265, 6.20681526904542088835730053039, 7.46261028773657489832215471574, 8.455181921164028207665051683361, 9.327820441539330420804608673051, 10.04517050900761317536685250745, 11.19897971912574422599178738873, 12.10713864521281605598821868486

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