Properties

Label 2-15e2-1.1-c3-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s + 30·7-s + 45·8-s − 50·11-s + 20·13-s + 150·14-s + 89·16-s − 10·17-s − 44·19-s − 250·22-s + 120·23-s + 100·26-s + 510·28-s + 50·29-s + 108·31-s + 85·32-s − 50·34-s + 40·37-s − 220·38-s − 400·41-s − 280·43-s − 850·44-s + 600·46-s − 280·47-s + 557·49-s + 340·52-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s + 1.61·7-s + 1.98·8-s − 1.37·11-s + 0.426·13-s + 2.86·14-s + 1.39·16-s − 0.142·17-s − 0.531·19-s − 2.42·22-s + 1.08·23-s + 0.754·26-s + 3.44·28-s + 0.320·29-s + 0.625·31-s + 0.469·32-s − 0.252·34-s + 0.177·37-s − 0.939·38-s − 1.52·41-s − 0.993·43-s − 2.91·44-s + 1.92·46-s − 0.868·47-s + 1.62·49-s + 0.906·52-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.384938520\)
\(L(\frac12)\) \(\approx\) \(5.384938520\)
\(L(\frac{5}{2})\) 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 \( 1 - 5 T + p^{3} T^{2} \)
7 \( 1 - 30 T + p^{3} T^{2} \)
11 \( 1 + 50 T + p^{3} T^{2} \)
13 \( 1 - 20 T + p^{3} T^{2} \)
17 \( 1 + 10 T + p^{3} T^{2} \)
19 \( 1 + 44 T + p^{3} T^{2} \)
23 \( 1 - 120 T + p^{3} T^{2} \)
29 \( 1 - 50 T + p^{3} T^{2} \)
31 \( 1 - 108 T + p^{3} T^{2} \)
37 \( 1 - 40 T + p^{3} T^{2} \)
41 \( 1 + 400 T + p^{3} T^{2} \)
43 \( 1 + 280 T + p^{3} T^{2} \)
47 \( 1 + 280 T + p^{3} T^{2} \)
53 \( 1 + 610 T + p^{3} T^{2} \)
59 \( 1 + 50 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 - 180 T + p^{3} T^{2} \)
71 \( 1 + 700 T + p^{3} T^{2} \)
73 \( 1 - 410 T + p^{3} T^{2} \)
79 \( 1 + 516 T + p^{3} T^{2} \)
83 \( 1 - 660 T + p^{3} T^{2} \)
89 \( 1 - 1500 T + p^{3} T^{2} \)
97 \( 1 - 1630 T + p^{3} T^{2} \)
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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

−11.90022322309422468578340810213, −11.15451861988398994005433142108, −10.47228413220981143202101022298, −8.484983028250606401510505327565, −7.58394761605198093727258901891, −6.33550168103295751971626822380, −5.06143801480476493733903819753, −4.71338840509793701084136909274, −3.14629717651220037628653934831, −1.85132318806004659956457135009, 1.85132318806004659956457135009, 3.14629717651220037628653934831, 4.71338840509793701084136909274, 5.06143801480476493733903819753, 6.33550168103295751971626822380, 7.58394761605198093727258901891, 8.484983028250606401510505327565, 10.47228413220981143202101022298, 11.15451861988398994005433142108, 11.90022322309422468578340810213

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