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Properties

Label 2-15e2-1.1-c7-0-11
Degree $2$
Conductor $225$
Sign $1$
Analytic cond. $70.2866$
Root an. cond. $8.38371$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 128·4-s + 1.25e3·7-s − 1.26e4·13-s + 1.63e4·16-s + 4.30e4·19-s − 1.60e5·28-s − 3.31e5·31-s − 2.79e5·37-s + 4.09e5·43-s + 7.51e5·49-s + 1.61e6·52-s + 1.99e6·61-s − 2.09e6·64-s + 4.05e6·67-s + 6.27e6·73-s − 5.51e6·76-s + 8.76e6·79-s − 1.58e7·91-s + 1.75e7·97-s − 8.02e6·103-s + 2.67e7·109-s + 2.05e7·112-s + ⋯
L(s)  = 1  − 4-s + 1.38·7-s − 1.59·13-s + 16-s + 1.44·19-s − 1.38·28-s − 1.99·31-s − 0.907·37-s + 0.785·43-s + 0.912·49-s + 1.59·52-s + 1.12·61-s − 64-s + 1.64·67-s + 1.88·73-s − 1.44·76-s + 1.99·79-s − 2.20·91-s + 1.94·97-s − 0.723·103-s + 1.97·109-s + 1.38·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(70.2866\)
Root analytic conductor: \(8.38371\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.672544965\)
\(L(\frac12)\) \(\approx\) \(1.672544965\)
\(L(\frac{9}{2})\) 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 \( 1 + p^{7} T^{2} \)
7 \( 1 - 1255 T + p^{7} T^{2} \)
11 \( 1 + p^{7} T^{2} \)
13 \( 1 + 12605 T + p^{7} T^{2} \)
17 \( 1 + p^{7} T^{2} \)
19 \( 1 - 43091 T + p^{7} T^{2} \)
23 \( 1 + p^{7} T^{2} \)
29 \( 1 + p^{7} T^{2} \)
31 \( 1 + 331387 T + p^{7} T^{2} \)
37 \( 1 + 279710 T + p^{7} T^{2} \)
41 \( 1 + p^{7} T^{2} \)
43 \( 1 - 409495 T + p^{7} T^{2} \)
47 \( 1 + p^{7} T^{2} \)
53 \( 1 + p^{7} T^{2} \)
59 \( 1 + p^{7} T^{2} \)
61 \( 1 - 1998347 T + p^{7} T^{2} \)
67 \( 1 - 4058455 T + p^{7} T^{2} \)
71 \( 1 + p^{7} T^{2} \)
73 \( 1 - 6274810 T + p^{7} T^{2} \)
79 \( 1 - 8763044 T + p^{7} T^{2} \)
83 \( 1 + p^{7} T^{2} \)
89 \( 1 + p^{7} T^{2} \)
97 \( 1 - 17521555 T + p^{7} T^{2} \)
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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

−10.99843050144852370980181120826, −9.852957815454752252805753852197, −9.087067316046095081282621540862, −7.970656211554168462769432015843, −7.29201720048977324040212141281, −5.32365828149134613629774041817, −4.96218272446456237936666739288, −3.66562642480412426145853828502, −2.03243167146514061235406675266, −0.68744509686538829116542154625, 0.68744509686538829116542154625, 2.03243167146514061235406675266, 3.66562642480412426145853828502, 4.96218272446456237936666739288, 5.32365828149134613629774041817, 7.29201720048977324040212141281, 7.970656211554168462769432015843, 9.087067316046095081282621540862, 9.852957815454752252805753852197, 10.99843050144852370980181120826

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