Properties

Label 2-15e2-1.1-c9-0-22
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  + 22·2-s − 28·4-s + 5.98e3·7-s − 1.18e4·8-s + 1.46e4·11-s − 3.79e4·13-s + 1.31e5·14-s − 2.47e5·16-s − 4.41e5·17-s + 4.41e5·19-s + 3.22e5·22-s + 2.26e6·23-s − 8.33e5·26-s − 1.67e5·28-s + 1.04e6·29-s − 7.91e6·31-s + 6.48e5·32-s − 9.70e6·34-s + 2.09e7·37-s + 9.72e6·38-s − 1.32e7·41-s + 2.31e7·43-s − 4.10e5·44-s + 4.98e7·46-s − 1.38e7·47-s − 4.49e6·49-s + 1.06e6·52-s + ⋯
L(s)  = 1  + 0.972·2-s − 0.0546·4-s + 0.942·7-s − 1.02·8-s + 0.301·11-s − 0.368·13-s + 0.916·14-s − 0.942·16-s − 1.28·17-s + 0.777·19-s + 0.293·22-s + 1.68·23-s − 0.357·26-s − 0.0515·28-s + 0.275·29-s − 1.53·31-s + 0.109·32-s − 1.24·34-s + 1.84·37-s + 0.756·38-s − 0.734·41-s + 1.03·43-s − 0.0164·44-s + 1.64·46-s − 0.414·47-s − 0.111·49-s + 0.0201·52-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

\(L(5)\) \(\approx\) \(3.397766923\)
\(L(\frac12)\) \(\approx\) \(3.397766923\)
\(L(\frac{11}{2})\) 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 \( 1 - 11 p T + p^{9} T^{2} \)
7 \( 1 - 5988 T + p^{9} T^{2} \)
11 \( 1 - 14648 T + p^{9} T^{2} \)
13 \( 1 + 37906 T + p^{9} T^{2} \)
17 \( 1 + 441098 T + p^{9} T^{2} \)
19 \( 1 - 441820 T + p^{9} T^{2} \)
23 \( 1 - 2264136 T + p^{9} T^{2} \)
29 \( 1 - 1049350 T + p^{9} T^{2} \)
31 \( 1 + 7910568 T + p^{9} T^{2} \)
37 \( 1 - 20992558 T + p^{9} T^{2} \)
41 \( 1 + 13285562 T + p^{9} T^{2} \)
43 \( 1 - 23130764 T + p^{9} T^{2} \)
47 \( 1 + 13873688 T + p^{9} T^{2} \)
53 \( 1 + 57635174 T + p^{9} T^{2} \)
59 \( 1 - 32042120 T + p^{9} T^{2} \)
61 \( 1 - 110664022 T + p^{9} T^{2} \)
67 \( 1 - 118568268 T + p^{9} T^{2} \)
71 \( 1 + 276679712 T + p^{9} T^{2} \)
73 \( 1 - 264023294 T + p^{9} T^{2} \)
79 \( 1 - 448202760 T + p^{9} T^{2} \)
83 \( 1 - 851015796 T + p^{9} T^{2} \)
89 \( 1 + 189894930 T + p^{9} T^{2} \)
97 \( 1 - 1014149278 T + p^{9} T^{2} \)
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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

−11.04132962608677993376323271893, −9.474424959600484829627820516101, −8.767613709682235618225980289724, −7.51820523322104871670821084672, −6.38622050334743562378732553998, −5.15463483602347535530003713844, −4.60242960540045227812763274198, −3.43014211147143646490018577837, −2.20696276703659377990430974393, −0.75048401086805536431263100816, 0.75048401086805536431263100816, 2.20696276703659377990430974393, 3.43014211147143646490018577837, 4.60242960540045227812763274198, 5.15463483602347535530003713844, 6.38622050334743562378732553998, 7.51820523322104871670821084672, 8.767613709682235618225980289724, 9.474424959600484829627820516101, 11.04132962608677993376323271893

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