Properties

Label 2-2800-1.1-c1-0-22
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  + 2·3-s + 7-s + 9-s + 2·17-s + 2·19-s + 2·21-s + 8·23-s − 4·27-s + 2·29-s − 4·31-s + 6·37-s − 2·41-s + 8·43-s − 4·47-s + 49-s + 4·51-s + 10·53-s + 4·57-s − 6·59-s + 4·61-s + 63-s − 12·67-s + 16·69-s + 14·73-s + 8·79-s − 11·81-s + 6·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 0.485·17-s + 0.458·19-s + 0.436·21-s + 1.66·23-s − 0.769·27-s + 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.37·53-s + 0.529·57-s − 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s + 1.92·69-s + 1.63·73-s + 0.900·79-s − 1.22·81-s + 0.658·83-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

\(L(1)\) \(\approx\) \(3.017092709\)
\(L(\frac12)\) \(\approx\) \(3.017092709\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−9.016353831659563256316815976220, −7.955680497262477941126609567507, −7.59496919133107363658161608903, −6.68922441578017008264572726362, −5.62940121509393392778473313445, −4.86690703180998964190536046386, −3.83051098046556122685042241853, −3.06551026181718468415456617058, −2.29037174921539130023849707486, −1.08114356407297375974866877413, 1.08114356407297375974866877413, 2.29037174921539130023849707486, 3.06551026181718468415456617058, 3.83051098046556122685042241853, 4.86690703180998964190536046386, 5.62940121509393392778473313445, 6.68922441578017008264572726362, 7.59496919133107363658161608903, 7.955680497262477941126609567507, 9.016353831659563256316815976220

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