Properties

Label 2-15e2-1.1-c9-0-7
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $115.883$
Root an. cond. $10.7648$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 18·2-s − 188·4-s − 9.12e3·7-s − 1.26e4·8-s − 2.11e4·11-s − 3.12e4·13-s − 1.64e5·14-s − 1.30e5·16-s − 2.79e5·17-s + 1.44e5·19-s − 3.80e5·22-s − 1.76e6·23-s − 5.61e5·26-s + 1.71e6·28-s − 4.69e6·29-s − 3.69e5·31-s + 4.10e6·32-s − 5.02e6·34-s − 9.34e6·37-s + 2.59e6·38-s + 7.22e6·41-s + 2.31e7·43-s + 3.97e6·44-s − 3.17e7·46-s + 2.29e7·47-s + 4.29e7·49-s + 5.86e6·52-s + ⋯
L(s)  = 1  + 0.795·2-s − 0.367·4-s − 1.43·7-s − 1.08·8-s − 0.435·11-s − 0.303·13-s − 1.14·14-s − 0.497·16-s − 0.811·17-s + 0.253·19-s − 0.346·22-s − 1.31·23-s − 0.241·26-s + 0.527·28-s − 1.23·29-s − 0.0717·31-s + 0.691·32-s − 0.645·34-s − 0.819·37-s + 0.201·38-s + 0.399·41-s + 1.03·43-s + 0.159·44-s − 1.04·46-s + 0.686·47-s + 1.06·49-s + 0.111·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(115.883\)
Root analytic conductor: \(10.7648\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(0.7752949509\)
\(L(\frac12)\) \(\approx\) \(0.7752949509\)
\(L(\frac{11}{2})\) 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 \( 1 - 9 p T + p^{9} T^{2} \)
7 \( 1 + 1304 p T + p^{9} T^{2} \)
11 \( 1 + 21132 T + p^{9} T^{2} \)
13 \( 1 + 31214 T + p^{9} T^{2} \)
17 \( 1 + 279342 T + p^{9} T^{2} \)
19 \( 1 - 7580 p T + p^{9} T^{2} \)
23 \( 1 + 1763496 T + p^{9} T^{2} \)
29 \( 1 + 4692510 T + p^{9} T^{2} \)
31 \( 1 + 369088 T + p^{9} T^{2} \)
37 \( 1 + 9347078 T + p^{9} T^{2} \)
41 \( 1 - 7226838 T + p^{9} T^{2} \)
43 \( 1 - 23147476 T + p^{9} T^{2} \)
47 \( 1 - 22971888 T + p^{9} T^{2} \)
53 \( 1 - 78477174 T + p^{9} T^{2} \)
59 \( 1 - 20310660 T + p^{9} T^{2} \)
61 \( 1 + 179339938 T + p^{9} T^{2} \)
67 \( 1 + 274528388 T + p^{9} T^{2} \)
71 \( 1 - 36342648 T + p^{9} T^{2} \)
73 \( 1 - 247089526 T + p^{9} T^{2} \)
79 \( 1 - 191874800 T + p^{9} T^{2} \)
83 \( 1 + 276159276 T + p^{9} T^{2} \)
89 \( 1 - 678997350 T + p^{9} T^{2} \)
97 \( 1 - 567657502 T + p^{9} T^{2} \)
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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

−10.55258910838638739088304239619, −9.579459400017826197165063370807, −8.890694918032067283084152469494, −7.47232654273064431598562071855, −6.29323212542301274348644517415, −5.54613685921410232922948221164, −4.27679820821117877936446225219, −3.41930163089060029521260871677, −2.35378438239309035674838055212, −0.34262773721765980781630651996, 0.34262773721765980781630651996, 2.35378438239309035674838055212, 3.41930163089060029521260871677, 4.27679820821117877936446225219, 5.54613685921410232922948221164, 6.29323212542301274348644517415, 7.47232654273064431598562071855, 8.890694918032067283084152469494, 9.579459400017826197165063370807, 10.55258910838638739088304239619

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