Properties

Label 2-15e2-1.1-c3-0-17
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 7·4-s + 24·7-s − 15·8-s − 52·11-s − 22·13-s + 24·14-s + 41·16-s − 14·17-s − 20·19-s − 52·22-s − 168·23-s − 22·26-s − 168·28-s − 230·29-s − 288·31-s + 161·32-s − 14·34-s + 34·37-s − 20·38-s − 122·41-s + 188·43-s + 364·44-s − 168·46-s + 256·47-s + 233·49-s + 154·52-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s + 1.29·7-s − 0.662·8-s − 1.42·11-s − 0.469·13-s + 0.458·14-s + 0.640·16-s − 0.199·17-s − 0.241·19-s − 0.503·22-s − 1.52·23-s − 0.165·26-s − 1.13·28-s − 1.47·29-s − 1.66·31-s + 0.889·32-s − 0.0706·34-s + 0.151·37-s − 0.0853·38-s − 0.464·41-s + 0.666·43-s + 1.24·44-s − 0.538·46-s + 0.794·47-s + 0.679·49-s + 0.410·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
7 \( 1 - 24 T + p^{3} T^{2} \)
11 \( 1 + 52 T + p^{3} T^{2} \)
13 \( 1 + 22 T + p^{3} T^{2} \)
17 \( 1 + 14 T + p^{3} T^{2} \)
19 \( 1 + 20 T + p^{3} T^{2} \)
23 \( 1 + 168 T + p^{3} T^{2} \)
29 \( 1 + 230 T + p^{3} T^{2} \)
31 \( 1 + 288 T + p^{3} T^{2} \)
37 \( 1 - 34 T + p^{3} T^{2} \)
41 \( 1 + 122 T + p^{3} T^{2} \)
43 \( 1 - 188 T + p^{3} T^{2} \)
47 \( 1 - 256 T + p^{3} T^{2} \)
53 \( 1 + 338 T + p^{3} T^{2} \)
59 \( 1 + 100 T + p^{3} T^{2} \)
61 \( 1 - 742 T + p^{3} T^{2} \)
67 \( 1 - 84 T + p^{3} T^{2} \)
71 \( 1 - 328 T + p^{3} T^{2} \)
73 \( 1 - 38 T + p^{3} T^{2} \)
79 \( 1 + 240 T + p^{3} T^{2} \)
83 \( 1 - 1212 T + p^{3} T^{2} \)
89 \( 1 + 330 T + p^{3} T^{2} \)
97 \( 1 + 866 T + p^{3} T^{2} \)
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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

−11.33036585079725910691291960735, −10.40986946477577123950040348774, −9.317510026974838406965733096606, −8.207668152509560229646308756247, −7.57311106360030147889706155959, −5.67678429626916430523903845128, −4.99767340874960418835195240187, −3.89304021135838981543981723556, −2.12634678512364983963856495165, 0, 2.12634678512364983963856495165, 3.89304021135838981543981723556, 4.99767340874960418835195240187, 5.67678429626916430523903845128, 7.57311106360030147889706155959, 8.207668152509560229646308756247, 9.317510026974838406965733096606, 10.40986946477577123950040348774, 11.33036585079725910691291960735

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