Properties

Label 2-15e2-1.1-c1-0-6
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 5·7-s − 5·13-s + 4·16-s − 19-s + 10·28-s − 7·31-s + 10·37-s − 5·43-s + 18·49-s + 10·52-s − 13·61-s − 8·64-s − 5·67-s + 10·73-s + 2·76-s − 4·79-s + 25·91-s − 5·97-s − 20·103-s − 19·109-s − 20·112-s + ⋯
L(s)  = 1  − 4-s − 1.88·7-s − 1.38·13-s + 16-s − 0.229·19-s + 1.88·28-s − 1.25·31-s + 1.64·37-s − 0.762·43-s + 18/7·49-s + 1.38·52-s − 1.66·61-s − 64-s − 0.610·67-s + 1.17·73-s + 0.229·76-s − 0.450·79-s + 2.62·91-s − 0.507·97-s − 1.97·103-s − 1.81·109-s − 1.88·112-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: \(1\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−12.11951399488920175322797663822, −10.49393585601286522315094653400, −9.599576350131101649309623215930, −9.206590206514279856778701903578, −7.75954149302713039814888795826, −6.62145632893118011347307968766, −5.46981299804941090347456591942, −4.14585724745469274814887485348, −2.93248946372060750659719660133, 0, 2.93248946372060750659719660133, 4.14585724745469274814887485348, 5.46981299804941090347456591942, 6.62145632893118011347307968766, 7.75954149302713039814888795826, 9.206590206514279856778701903578, 9.599576350131101649309623215930, 10.49393585601286522315094653400, 12.11951399488920175322797663822

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