Properties

Label 2-15e2-1.1-c11-0-70
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $172.877$
Root an. cond. $13.1482$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 34·2-s − 892·4-s + 1.75e4·7-s − 9.99e4·8-s + 4.68e5·11-s + 3.74e5·13-s + 5.96e5·14-s − 1.57e6·16-s − 3.72e6·17-s − 3.79e5·19-s + 1.59e7·22-s − 3.24e7·23-s + 1.27e7·26-s − 1.56e7·28-s − 6.96e7·29-s + 1.71e8·31-s + 1.51e8·32-s − 1.26e8·34-s + 2.91e8·37-s − 1.29e7·38-s − 1.91e8·41-s + 1.75e9·43-s − 4.18e8·44-s − 1.10e9·46-s + 1.62e9·47-s − 1.66e9·49-s − 3.33e8·52-s + ⋯
L(s)  = 1  + 0.751·2-s − 0.435·4-s + 0.394·7-s − 1.07·8-s + 0.877·11-s + 0.279·13-s + 0.296·14-s − 0.374·16-s − 0.636·17-s − 0.0351·19-s + 0.659·22-s − 1.05·23-s + 0.209·26-s − 0.171·28-s − 0.630·29-s + 1.07·31-s + 0.796·32-s − 0.477·34-s + 0.690·37-s − 0.0264·38-s − 0.257·41-s + 1.82·43-s − 0.382·44-s − 0.790·46-s + 1.03·47-s − 0.844·49-s − 0.121·52-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: yes
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 \( 1 - 17 p T + p^{11} T^{2} \)
7 \( 1 - 2508 p T + p^{11} T^{2} \)
11 \( 1 - 468788 T + p^{11} T^{2} \)
13 \( 1 - 374042 T + p^{11} T^{2} \)
17 \( 1 + 3724286 T + p^{11} T^{2} \)
19 \( 1 + 379460 T + p^{11} T^{2} \)
23 \( 1 + 32458092 T + p^{11} T^{2} \)
29 \( 1 + 69696710 T + p^{11} T^{2} \)
31 \( 1 - 171448632 T + p^{11} T^{2} \)
37 \( 1 - 291340546 T + p^{11} T^{2} \)
41 \( 1 + 191343242 T + p^{11} T^{2} \)
43 \( 1 - 1759857392 T + p^{11} T^{2} \)
47 \( 1 - 1623469924 T + p^{11} T^{2} \)
53 \( 1 + 644888642 T + p^{11} T^{2} \)
59 \( 1 + 925569220 T + p^{11} T^{2} \)
61 \( 1 + 10898589338 T + p^{11} T^{2} \)
67 \( 1 + 3795674064 T + p^{11} T^{2} \)
71 \( 1 - 22966943728 T + p^{11} T^{2} \)
73 \( 1 + 9880820458 T + p^{11} T^{2} \)
79 \( 1 + 20768886240 T + p^{11} T^{2} \)
83 \( 1 - 3204862008 T + p^{11} T^{2} \)
89 \( 1 + 63176321130 T + p^{11} T^{2} \)
97 \( 1 + 126494473874 T + p^{11} T^{2} \)
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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

−9.654545631614378729517185210299, −8.886922918709171364624397535600, −7.892823322800810683718924780841, −6.49753025694724846106933919702, −5.70255636711372028562311505827, −4.48202430106997520977059397360, −3.91327848022365141866084606521, −2.61470968164577575293272677894, −1.26127240094190220718269134434, 0, 1.26127240094190220718269134434, 2.61470968164577575293272677894, 3.91327848022365141866084606521, 4.48202430106997520977059397360, 5.70255636711372028562311505827, 6.49753025694724846106933919702, 7.892823322800810683718924780841, 8.886922918709171364624397535600, 9.654545631614378729517185210299

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