Properties

Label 2-15e2-1.1-c7-0-43
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·2-s − 92·4-s + 64·7-s − 1.32e3·8-s + 948·11-s + 5.09e3·13-s + 384·14-s + 3.85e3·16-s + 2.83e4·17-s − 8.62e3·19-s + 5.68e3·22-s − 1.52e4·23-s + 3.05e4·26-s − 5.88e3·28-s − 3.65e4·29-s − 2.76e5·31-s + 1.92e5·32-s + 1.70e5·34-s − 2.68e5·37-s − 5.17e4·38-s + 6.29e5·41-s − 6.85e5·43-s − 8.72e4·44-s − 9.17e4·46-s + 5.83e5·47-s − 8.19e5·49-s − 4.69e5·52-s + ⋯
L(s)  = 1  + 0.530·2-s − 0.718·4-s + 0.0705·7-s − 0.911·8-s + 0.214·11-s + 0.643·13-s + 0.0374·14-s + 0.235·16-s + 1.40·17-s − 0.288·19-s + 0.113·22-s − 0.262·23-s + 0.341·26-s − 0.0506·28-s − 0.277·29-s − 1.66·31-s + 1.03·32-s + 0.743·34-s − 0.871·37-s − 0.152·38-s + 1.42·41-s − 1.31·43-s − 0.154·44-s − 0.138·46-s + 0.819·47-s − 0.995·49-s − 0.462·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 3 p T + p^{7} T^{2} \)
7 \( 1 - 64 T + p^{7} T^{2} \)
11 \( 1 - 948 T + p^{7} T^{2} \)
13 \( 1 - 5098 T + p^{7} T^{2} \)
17 \( 1 - 28386 T + p^{7} T^{2} \)
19 \( 1 + 8620 T + p^{7} T^{2} \)
23 \( 1 + 15288 T + p^{7} T^{2} \)
29 \( 1 + 36510 T + p^{7} T^{2} \)
31 \( 1 + 276808 T + p^{7} T^{2} \)
37 \( 1 + 268526 T + p^{7} T^{2} \)
41 \( 1 - 629718 T + p^{7} T^{2} \)
43 \( 1 + 685772 T + p^{7} T^{2} \)
47 \( 1 - 583296 T + p^{7} T^{2} \)
53 \( 1 + 428058 T + p^{7} T^{2} \)
59 \( 1 + 1306380 T + p^{7} T^{2} \)
61 \( 1 - 300662 T + p^{7} T^{2} \)
67 \( 1 - 507244 T + p^{7} T^{2} \)
71 \( 1 + 5560632 T + p^{7} T^{2} \)
73 \( 1 + 1369082 T + p^{7} T^{2} \)
79 \( 1 + 6913720 T + p^{7} T^{2} \)
83 \( 1 + 4376748 T + p^{7} T^{2} \)
89 \( 1 - 8528310 T + p^{7} T^{2} \)
97 \( 1 - 8826814 T + p^{7} T^{2} \)
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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

−10.48220739083952401567796680959, −9.454575366419476483621309686856, −8.605112063627316628366701710250, −7.53118232572431737988300703368, −6.09154458004630527627844170294, −5.28656742260754299662132358906, −4.06602977757791761397196072567, −3.20062292759901484798136305874, −1.42301116654435969034221949571, 0, 1.42301116654435969034221949571, 3.20062292759901484798136305874, 4.06602977757791761397196072567, 5.28656742260754299662132358906, 6.09154458004630527627844170294, 7.53118232572431737988300703368, 8.605112063627316628366701710250, 9.454575366419476483621309686856, 10.48220739083952401567796680959

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