Properties

Label 2-2800-1.1-c1-0-18
Degree $2$
Conductor $2800$
Sign $1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
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  − 3·3-s + 7-s + 6·9-s + 5·11-s + 5·13-s + 7·17-s + 2·19-s − 3·21-s − 2·23-s − 9·27-s + 7·29-s − 4·31-s − 15·33-s + 6·37-s − 15·39-s − 12·41-s − 2·43-s + 47-s + 49-s − 21·51-s − 6·57-s + 4·59-s + 4·61-s + 6·63-s + 8·67-s + 6·69-s − 6·73-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.377·7-s + 2·9-s + 1.50·11-s + 1.38·13-s + 1.69·17-s + 0.458·19-s − 0.654·21-s − 0.417·23-s − 1.73·27-s + 1.29·29-s − 0.718·31-s − 2.61·33-s + 0.986·37-s − 2.40·39-s − 1.87·41-s − 0.304·43-s + 0.145·47-s + 1/7·49-s − 2.94·51-s − 0.794·57-s + 0.520·59-s + 0.512·61-s + 0.755·63-s + 0.977·67-s + 0.722·69-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.418753993\)
\(L(\frac12)\) \(\approx\) \(1.418753993\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.795758213468663956094183111092, −7.984482054980256459193096634432, −6.97670663684812382164401826864, −6.39460459437087758891198231048, −5.76481244688009992103280329559, −5.12180714528748314961399662749, −4.15041618199622899610530288269, −3.45165323807794852534236235819, −1.47823230313481939170754494140, −0.949301332029487315300827868877, 0.949301332029487315300827868877, 1.47823230313481939170754494140, 3.45165323807794852534236235819, 4.15041618199622899610530288269, 5.12180714528748314961399662749, 5.76481244688009992103280329559, 6.39460459437087758891198231048, 6.97670663684812382164401826864, 7.984482054980256459193096634432, 8.795758213468663956094183111092

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