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Properties

Label 2-1512-1.1-c1-0-22
Degree $2$
Conductor $1512$
Sign $-1$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 6·13-s − 7·17-s − 8·19-s − 2·23-s − 4·25-s + 4·29-s − 2·31-s + 35-s + 9·37-s + 11·41-s − 7·43-s − 11·47-s + 49-s + 10·53-s + 9·59-s − 8·61-s − 6·65-s − 12·71-s − 14·73-s + 79-s − 9·83-s − 7·85-s + 14·89-s − 6·91-s − 8·95-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 1.66·13-s − 1.69·17-s − 1.83·19-s − 0.417·23-s − 4/5·25-s + 0.742·29-s − 0.359·31-s + 0.169·35-s + 1.47·37-s + 1.71·41-s − 1.06·43-s − 1.60·47-s + 1/7·49-s + 1.37·53-s + 1.17·59-s − 1.02·61-s − 0.744·65-s − 1.42·71-s − 1.63·73-s + 0.112·79-s − 0.987·83-s − 0.759·85-s + 1.48·89-s − 0.628·91-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-1$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−9.076883100182159314171142406491, −8.335845467804548813626378587065, −7.45727589406932757057223737571, −6.59678521703246191744427531722, −5.87670198471138754367852660011, −4.67556818293723745978475387166, −4.27170572622829685531677010666, −2.56550471034343739528833531441, −1.99776993177936008547630863670, 0, 1.99776993177936008547630863670, 2.56550471034343739528833531441, 4.27170572622829685531677010666, 4.67556818293723745978475387166, 5.87670198471138754367852660011, 6.59678521703246191744427531722, 7.45727589406932757057223737571, 8.335845467804548813626378587065, 9.076883100182159314171142406491

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