Properties

Label 2-42e2-1.1-c1-0-9
Degree $2$
Conductor $1764$
Sign $1$
Analytic cond. $14.0856$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 2·11-s + 6·13-s − 4·17-s + 4·19-s − 2·23-s + 11·25-s + 2·29-s + 2·37-s − 4·43-s + 12·47-s + 6·53-s − 8·55-s − 8·59-s − 6·61-s + 24·65-s − 8·67-s − 14·71-s + 2·73-s + 12·79-s − 4·83-s − 16·85-s + 16·95-s + 2·97-s + 16·101-s + 16·103-s − 18·107-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.603·11-s + 1.66·13-s − 0.970·17-s + 0.917·19-s − 0.417·23-s + 11/5·25-s + 0.371·29-s + 0.328·37-s − 0.609·43-s + 1.75·47-s + 0.824·53-s − 1.07·55-s − 1.04·59-s − 0.768·61-s + 2.97·65-s − 0.977·67-s − 1.66·71-s + 0.234·73-s + 1.35·79-s − 0.439·83-s − 1.73·85-s + 1.64·95-s + 0.203·97-s + 1.59·101-s + 1.57·103-s − 1.74·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(14.0856\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1764} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1764,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.546503206\)
\(L(\frac12)\) \(\approx\) \(2.546503206\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.133591619397453241615790856366, −8.847117284114771599871123870971, −7.74475735714765512283633749961, −6.66397314528471112905733886770, −6.01041095937836104246896195058, −5.47944807329373236781153563032, −4.44067514856766940885014862243, −3.15949197761990841887540386198, −2.19665073610568722129910000758, −1.21434687808983120631223068182, 1.21434687808983120631223068182, 2.19665073610568722129910000758, 3.15949197761990841887540386198, 4.44067514856766940885014862243, 5.47944807329373236781153563032, 6.01041095937836104246896195058, 6.66397314528471112905733886770, 7.74475735714765512283633749961, 8.847117284114771599871123870971, 9.133591619397453241615790856366

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