Properties

Label 2-15e2-1.1-c9-0-34
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  − 4·2-s − 496·4-s + 7.68e3·7-s + 4.03e3·8-s + 8.64e4·11-s + 1.49e5·13-s − 3.07e4·14-s + 2.37e5·16-s − 2.07e5·17-s + 7.16e5·19-s − 3.45e5·22-s + 1.36e6·23-s − 5.99e5·26-s − 3.80e6·28-s + 3.19e6·29-s − 2.34e6·31-s − 3.01e6·32-s + 8.30e5·34-s − 1.87e7·37-s − 2.86e6·38-s + 2.92e7·41-s + 1.51e6·43-s − 4.28e7·44-s − 5.47e6·46-s + 6.15e5·47-s + 1.86e7·49-s − 7.43e7·52-s + ⋯
L(s)  = 1  − 0.176·2-s − 0.968·4-s + 1.20·7-s + 0.348·8-s + 1.77·11-s + 1.45·13-s − 0.213·14-s + 0.907·16-s − 0.602·17-s + 1.26·19-s − 0.314·22-s + 1.02·23-s − 0.257·26-s − 1.17·28-s + 0.838·29-s − 0.456·31-s − 0.508·32-s + 0.106·34-s − 1.64·37-s − 0.222·38-s + 1.61·41-s + 0.0676·43-s − 1.72·44-s − 0.180·46-s + 0.0184·47-s + 0.461·49-s − 1.41·52-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

\(L(5)\) \(\approx\) \(2.667231937\)
\(L(\frac12)\) \(\approx\) \(2.667231937\)
\(L(\frac{11}{2})\) 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 \( 1 + p^{2} T + p^{9} T^{2} \)
7 \( 1 - 7680 T + p^{9} T^{2} \)
11 \( 1 - 86404 T + p^{9} T^{2} \)
13 \( 1 - 149978 T + p^{9} T^{2} \)
17 \( 1 + 207622 T + p^{9} T^{2} \)
19 \( 1 - 716284 T + p^{9} T^{2} \)
23 \( 1 - 1369920 T + p^{9} T^{2} \)
29 \( 1 - 3194402 T + p^{9} T^{2} \)
31 \( 1 + 2349000 T + p^{9} T^{2} \)
37 \( 1 + 18735710 T + p^{9} T^{2} \)
41 \( 1 - 29282630 T + p^{9} T^{2} \)
43 \( 1 - 1516724 T + p^{9} T^{2} \)
47 \( 1 - 615752 T + p^{9} T^{2} \)
53 \( 1 - 4747430 T + p^{9} T^{2} \)
59 \( 1 + 60616076 T + p^{9} T^{2} \)
61 \( 1 + 126745682 T + p^{9} T^{2} \)
67 \( 1 - 111182652 T + p^{9} T^{2} \)
71 \( 1 - 175551608 T + p^{9} T^{2} \)
73 \( 1 - 61233350 T + p^{9} T^{2} \)
79 \( 1 - 234431160 T + p^{9} T^{2} \)
83 \( 1 - 118910388 T + p^{9} T^{2} \)
89 \( 1 - 316534326 T + p^{9} T^{2} \)
97 \( 1 + 242912258 T + p^{9} T^{2} \)
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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

−10.73694016710801903987370158173, −9.260313664792810891288802620079, −8.866440864594454659394416031351, −7.88424166218632988662658806801, −6.62205688477745553992429577914, −5.35944804921787368015338609729, −4.36587054277966937156581317084, −3.50039231010328262300823595975, −1.47469768411427914316117936664, −0.946182352267444674550513754532, 0.946182352267444674550513754532, 1.47469768411427914316117936664, 3.50039231010328262300823595975, 4.36587054277966937156581317084, 5.35944804921787368015338609729, 6.62205688477745553992429577914, 7.88424166218632988662658806801, 8.866440864594454659394416031351, 9.260313664792810891288802620079, 10.73694016710801903987370158173

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