Properties

Label 2-15e2-1.1-c3-0-5
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  + 2-s − 7·4-s − 6·7-s − 15·8-s + 43·11-s + 28·13-s − 6·14-s + 41·16-s + 91·17-s − 35·19-s + 43·22-s + 162·23-s + 28·26-s + 42·28-s − 160·29-s + 42·31-s + 161·32-s + 91·34-s + 314·37-s − 35·38-s + 203·41-s − 92·43-s − 301·44-s + 162·46-s + 196·47-s − 307·49-s − 196·52-s + ⋯
L(s)  = 1  + 0.353·2-s − 7/8·4-s − 0.323·7-s − 0.662·8-s + 1.17·11-s + 0.597·13-s − 0.114·14-s + 0.640·16-s + 1.29·17-s − 0.422·19-s + 0.416·22-s + 1.46·23-s + 0.211·26-s + 0.283·28-s − 1.02·29-s + 0.243·31-s + 0.889·32-s + 0.459·34-s + 1.39·37-s − 0.149·38-s + 0.773·41-s − 0.326·43-s − 1.03·44-s + 0.519·46-s + 0.608·47-s − 0.895·49-s − 0.522·52-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

\(L(2)\) \(\approx\) \(1.758102196\)
\(L(\frac12)\) \(\approx\) \(1.758102196\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
7 \( 1 + 6 T + p^{3} T^{2} \)
11 \( 1 - 43 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 - 91 T + p^{3} T^{2} \)
19 \( 1 + 35 T + p^{3} T^{2} \)
23 \( 1 - 162 T + p^{3} T^{2} \)
29 \( 1 + 160 T + p^{3} T^{2} \)
31 \( 1 - 42 T + p^{3} T^{2} \)
37 \( 1 - 314 T + p^{3} T^{2} \)
41 \( 1 - 203 T + p^{3} T^{2} \)
43 \( 1 + 92 T + p^{3} T^{2} \)
47 \( 1 - 196 T + p^{3} T^{2} \)
53 \( 1 - 82 T + p^{3} T^{2} \)
59 \( 1 - 280 T + p^{3} T^{2} \)
61 \( 1 + 518 T + p^{3} T^{2} \)
67 \( 1 + 141 T + p^{3} T^{2} \)
71 \( 1 + 412 T + p^{3} T^{2} \)
73 \( 1 - 763 T + p^{3} T^{2} \)
79 \( 1 - 510 T + p^{3} T^{2} \)
83 \( 1 - 777 T + p^{3} T^{2} \)
89 \( 1 - 945 T + p^{3} T^{2} \)
97 \( 1 + 1246 T + p^{3} T^{2} \)
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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

−11.98730970536149473236317279420, −10.89791083716000329796113314780, −9.595262671887968331310211224728, −9.058656936127540880872239179374, −7.893931738396455026951832238952, −6.50273023718883958193675517949, −5.50551355875410980607343205567, −4.22013956890369447585937434634, −3.26085699027788409524718115330, −1.00572876630883561368004944942, 1.00572876630883561368004944942, 3.26085699027788409524718115330, 4.22013956890369447585937434634, 5.50551355875410980607343205567, 6.50273023718883958193675517949, 7.893931738396455026951832238952, 9.058656936127540880872239179374, 9.595262671887968331310211224728, 10.89791083716000329796113314780, 11.98730970536149473236317279420

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