Properties

Label 2-15e2-1.1-c3-0-9
Degree $2$
Conductor $225$
Sign $-1$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 8·4-s − 6·7-s − 32·11-s + 38·13-s + 24·14-s − 64·16-s + 26·17-s + 100·19-s + 128·22-s − 78·23-s − 152·26-s − 48·28-s + 50·29-s − 108·31-s + 256·32-s − 104·34-s − 266·37-s − 400·38-s − 22·41-s − 442·43-s − 256·44-s + 312·46-s − 514·47-s − 307·49-s + 304·52-s + 2·53-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.323·7-s − 0.877·11-s + 0.810·13-s + 0.458·14-s − 16-s + 0.370·17-s + 1.20·19-s + 1.24·22-s − 0.707·23-s − 1.14·26-s − 0.323·28-s + 0.320·29-s − 0.625·31-s + 1.41·32-s − 0.524·34-s − 1.18·37-s − 1.70·38-s − 0.0838·41-s − 1.56·43-s − 0.877·44-s + 1.00·46-s − 1.59·47-s − 0.895·49-s + 0.810·52-s + 0.00518·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p^{2} T + p^{3} T^{2} \)
7 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 + 32 T + p^{3} T^{2} \)
13 \( 1 - 38 T + p^{3} T^{2} \)
17 \( 1 - 26 T + p^{3} T^{2} \)
19 \( 1 - 100 T + p^{3} T^{2} \)
23 \( 1 + 78 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 + 108 T + p^{3} T^{2} \)
37 \( 1 + 266 T + p^{3} T^{2} \)
41 \( 1 + 22 T + p^{3} T^{2} \)
43 \( 1 + 442 T + p^{3} T^{2} \)
47 \( 1 + 514 T + p^{3} T^{2} \)
53 \( 1 - 2 T + p^{3} T^{2} \)
59 \( 1 + 500 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 + 126 T + p^{3} T^{2} \)
71 \( 1 + 412 T + p^{3} T^{2} \)
73 \( 1 - 878 T + p^{3} T^{2} \)
79 \( 1 - 600 T + p^{3} T^{2} \)
83 \( 1 - 282 T + p^{3} T^{2} \)
89 \( 1 - 150 T + p^{3} T^{2} \)
97 \( 1 + 386 T + p^{3} T^{2} \)
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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

−10.96926519078076475361092384039, −10.15326347085816014270893512129, −9.413637005742010778557720526341, −8.344434913247458594535211323479, −7.65281470367648967595994536620, −6.49839143912079036043922633202, −5.10283943943181689170567094516, −3.29610350142201987915407042625, −1.57425675175983214824376213776, 0, 1.57425675175983214824376213776, 3.29610350142201987915407042625, 5.10283943943181689170567094516, 6.49839143912079036043922633202, 7.65281470367648967595994536620, 8.344434913247458594535211323479, 9.413637005742010778557720526341, 10.15326347085816014270893512129, 10.96926519078076475361092384039

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