Properties

Label 2-15e2-1.1-c3-0-6
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: $\chi_{225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.013168598\)
\(L(\frac12)\) \(\approx\) \(1.013168598\)
\(L(\frac{5}{2})\) 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

−11.44278418795499504354227826887, −10.76864782583385601868773747727, −9.708932457997438284766225238529, −8.716161092747400496099370407717, −8.143490742073998796989921567154, −7.15455693809408349084712741806, −5.98991984538475014430491883109, −4.21987352485402673603629213990, −2.05367613267347723198425356008, −1.05774706524711401719953831556, 1.05774706524711401719953831556, 2.05367613267347723198425356008, 4.21987352485402673603629213990, 5.98991984538475014430491883109, 7.15455693809408349084712741806, 8.143490742073998796989921567154, 8.716161092747400496099370407717, 9.708932457997438284766225238529, 10.76864782583385601868773747727, 11.44278418795499504354227826887

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