Properties

Label 2-15e2-1.1-c5-0-8
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·2-s + 4·4-s + 40·7-s + 168·8-s + 564·11-s − 638·13-s − 240·14-s − 1.13e3·16-s + 882·17-s − 556·19-s − 3.38e3·22-s − 840·23-s + 3.82e3·26-s + 160·28-s − 4.63e3·29-s + 4.40e3·31-s + 1.44e3·32-s − 5.29e3·34-s + 2.41e3·37-s + 3.33e3·38-s + 6.87e3·41-s − 9.64e3·43-s + 2.25e3·44-s + 5.04e3·46-s − 1.86e4·47-s − 1.52e4·49-s − 2.55e3·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s + 0.308·7-s + 0.928·8-s + 1.40·11-s − 1.04·13-s − 0.327·14-s − 1.10·16-s + 0.740·17-s − 0.353·19-s − 1.49·22-s − 0.331·23-s + 1.11·26-s + 0.0385·28-s − 1.02·29-s + 0.822·31-s + 0.248·32-s − 0.785·34-s + 0.289·37-s + 0.374·38-s + 0.638·41-s − 0.795·43-s + 0.175·44-s + 0.351·46-s − 1.23·47-s − 0.904·49-s − 0.130·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.043876569\)
\(L(\frac12)\) \(\approx\) \(1.043876569\)
\(L(\frac{7}{2})\) 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 \( 1 + 3 p T + p^{5} T^{2} \)
7 \( 1 - 40 T + p^{5} T^{2} \)
11 \( 1 - 564 T + p^{5} T^{2} \)
13 \( 1 + 638 T + p^{5} T^{2} \)
17 \( 1 - 882 T + p^{5} T^{2} \)
19 \( 1 + 556 T + p^{5} T^{2} \)
23 \( 1 + 840 T + p^{5} T^{2} \)
29 \( 1 + 4638 T + p^{5} T^{2} \)
31 \( 1 - 4400 T + p^{5} T^{2} \)
37 \( 1 - 2410 T + p^{5} T^{2} \)
41 \( 1 - 6870 T + p^{5} T^{2} \)
43 \( 1 + 9644 T + p^{5} T^{2} \)
47 \( 1 + 18672 T + p^{5} T^{2} \)
53 \( 1 - 33750 T + p^{5} T^{2} \)
59 \( 1 - 18084 T + p^{5} T^{2} \)
61 \( 1 - 39758 T + p^{5} T^{2} \)
67 \( 1 - 23068 T + p^{5} T^{2} \)
71 \( 1 - 4248 T + p^{5} T^{2} \)
73 \( 1 - 41110 T + p^{5} T^{2} \)
79 \( 1 - 21920 T + p^{5} T^{2} \)
83 \( 1 - 82452 T + p^{5} T^{2} \)
89 \( 1 - 94086 T + p^{5} T^{2} \)
97 \( 1 + 49442 T + p^{5} T^{2} \)
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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

−11.26977968289901936934165739949, −10.04451733655426074069957083210, −9.499991932985028341312008685745, −8.496963485501366448185058489528, −7.61562418678788040685984990326, −6.58836163513160959831586206517, −5.04764557081578273808900281220, −3.87193174819542469752503566883, −1.97082124546754790830917560066, −0.74240855904815400054244859366, 0.74240855904815400054244859366, 1.97082124546754790830917560066, 3.87193174819542469752503566883, 5.04764557081578273808900281220, 6.58836163513160959831586206517, 7.61562418678788040685984990326, 8.496963485501366448185058489528, 9.499991932985028341312008685745, 10.04451733655426074069957083210, 11.26977968289901936934165739949

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