Properties

Label 2-15e2-1.1-c11-0-48
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  − 56·2-s + 1.08e3·4-s − 2.79e4·7-s + 5.37e4·8-s + 1.12e5·11-s + 1.09e6·13-s + 1.56e6·14-s − 5.23e6·16-s − 2.49e5·17-s − 1.37e7·19-s − 6.27e6·22-s + 4.13e7·23-s − 6.14e7·26-s − 3.04e7·28-s + 4.53e6·29-s − 2.65e8·31-s + 1.83e8·32-s + 1.39e7·34-s + 2.12e8·37-s + 7.67e8·38-s + 1.26e9·41-s − 1.41e7·43-s + 1.21e8·44-s − 2.31e9·46-s − 2.65e9·47-s − 1.19e9·49-s + 1.19e9·52-s + ⋯
L(s)  = 1  − 1.23·2-s + 0.531·4-s − 0.629·7-s + 0.580·8-s + 0.209·11-s + 0.819·13-s + 0.778·14-s − 1.24·16-s − 0.0426·17-s − 1.27·19-s − 0.259·22-s + 1.34·23-s − 1.01·26-s − 0.334·28-s + 0.0410·29-s − 1.66·31-s + 0.965·32-s + 0.0527·34-s + 0.502·37-s + 1.57·38-s + 1.70·41-s − 0.0146·43-s + 0.111·44-s − 1.65·46-s − 1.69·47-s − 0.603·49-s + 0.435·52-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 \( 1 + 7 p^{3} T + p^{11} T^{2} \)
7 \( 1 + 27984 T + p^{11} T^{2} \)
11 \( 1 - 112028 T + p^{11} T^{2} \)
13 \( 1 - 1096922 T + p^{11} T^{2} \)
17 \( 1 + 249566 T + p^{11} T^{2} \)
19 \( 1 + 13712420 T + p^{11} T^{2} \)
23 \( 1 - 41395728 T + p^{11} T^{2} \)
29 \( 1 - 4533850 T + p^{11} T^{2} \)
31 \( 1 + 265339008 T + p^{11} T^{2} \)
37 \( 1 - 212136946 T + p^{11} T^{2} \)
41 \( 1 - 1266969958 T + p^{11} T^{2} \)
43 \( 1 + 14129548 T + p^{11} T^{2} \)
47 \( 1 + 2657273336 T + p^{11} T^{2} \)
53 \( 1 - 2402699278 T + p^{11} T^{2} \)
59 \( 1 + 7498737220 T + p^{11} T^{2} \)
61 \( 1 + 4064828858 T + p^{11} T^{2} \)
67 \( 1 + 6871514244 T + p^{11} T^{2} \)
71 \( 1 - 13283734648 T + p^{11} T^{2} \)
73 \( 1 - 28875844262 T + p^{11} T^{2} \)
79 \( 1 - 27100302240 T + p^{11} T^{2} \)
83 \( 1 + 34365255132 T + p^{11} T^{2} \)
89 \( 1 - 63500412630 T + p^{11} T^{2} \)
97 \( 1 + 19634495234 T + p^{11} T^{2} \)
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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

−9.518874368195398620057267601270, −8.993670653611198819098636100098, −8.064060319741354000466271401039, −7.01807048825565172754043340462, −6.11676475808548831608347403094, −4.63278819203964940088670148527, −3.44645655229117234642368564267, −2.04394094918571894564906394145, −0.972107214510196005488072331044, 0, 0.972107214510196005488072331044, 2.04394094918571894564906394145, 3.44645655229117234642368564267, 4.63278819203964940088670148527, 6.11676475808548831608347403094, 7.01807048825565172754043340462, 8.064060319741354000466271401039, 8.993670653611198819098636100098, 9.518874368195398620057267601270

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