Properties

Label 2-1428-1.1-c1-0-2
Degree $2$
Conductor $1428$
Sign $1$
Analytic cond. $11.4026$
Root an. cond. $3.37677$
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  − 3-s + 2·5-s − 7-s + 9-s + 6·13-s − 2·15-s − 17-s − 2·19-s + 21-s − 25-s − 27-s + 8·29-s − 2·35-s + 2·37-s − 6·39-s + 2·41-s + 8·43-s + 2·45-s − 8·47-s + 49-s + 51-s − 2·53-s + 2·57-s + 12·59-s − 4·61-s − 63-s + 12·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.66·13-s − 0.516·15-s − 0.242·17-s − 0.458·19-s + 0.218·21-s − 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.338·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.140·51-s − 0.274·53-s + 0.264·57-s + 1.56·59-s − 0.512·61-s − 0.125·63-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1428\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(11.4026\)
Root analytic conductor: \(3.37677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1428} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1428,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.671110722\)
\(L(\frac12)\) \(\approx\) \(1.671110722\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−9.627694822849684257986488931214, −8.810858271674775165743259094528, −8.034818988611744078902949562023, −6.74786322217944422871622167174, −6.23937732241094617204326244710, −5.61803835729553467448193127024, −4.52714021066437646502043575114, −3.52683087005780987139144003068, −2.23959892983332666617681545788, −1.00570474672251756736233046600, 1.00570474672251756736233046600, 2.23959892983332666617681545788, 3.52683087005780987139144003068, 4.52714021066437646502043575114, 5.61803835729553467448193127024, 6.23937732241094617204326244710, 6.74786322217944422871622167174, 8.034818988611744078902949562023, 8.810858271674775165743259094528, 9.627694822849684257986488931214

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