Properties

Label 2-8400-1.1-c1-0-56
Degree $2$
Conductor $8400$
Sign $1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s + 3.80·11-s − 0.622·13-s + 4.42·17-s − 0.622·19-s + 21-s + 2.62·23-s + 27-s + 9.61·29-s + 0.622·31-s + 3.80·33-s + 1.24·37-s − 0.622·39-s + 4.62·41-s − 4.85·43-s + 11.6·47-s + 49-s + 4.42·51-s − 13.4·53-s − 0.622·57-s − 11.6·59-s − 8.10·61-s + 63-s + 2.62·69-s − 2.56·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.14·11-s − 0.172·13-s + 1.07·17-s − 0.142·19-s + 0.218·21-s + 0.546·23-s + 0.192·27-s + 1.78·29-s + 0.111·31-s + 0.662·33-s + 0.204·37-s − 0.0996·39-s + 0.721·41-s − 0.740·43-s + 1.69·47-s + 0.142·49-s + 0.620·51-s − 1.85·53-s − 0.0824·57-s − 1.51·59-s − 1.03·61-s + 0.125·63-s + 0.315·69-s − 0.304·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8400\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(67.0743\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.346451932\)
\(L(\frac12)\) \(\approx\) \(3.346451932\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 - 3.80T + 11T^{2} \)
13 \( 1 + 0.622T + 13T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 + 0.622T + 19T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 - 9.61T + 29T^{2} \)
31 \( 1 - 0.622T + 31T^{2} \)
37 \( 1 - 1.24T + 37T^{2} \)
41 \( 1 - 4.62T + 41T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 8.10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 6.75T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + 8.23T + 89T^{2} \)
97 \( 1 + 4.23T + 97T^{2} \)
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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

−7.85347257344309858946451212904, −7.19444929522728633540087632574, −6.45423508559015539058118507184, −5.82828512353959628894767772970, −4.78416831151292123118871160746, −4.33507909732403557736725737123, −3.37139725257143495796766936367, −2.79295555995698333158212282310, −1.67182960224454113879544873335, −0.949326910976761379206311139407, 0.949326910976761379206311139407, 1.67182960224454113879544873335, 2.79295555995698333158212282310, 3.37139725257143495796766936367, 4.33507909732403557736725737123, 4.78416831151292123118871160746, 5.82828512353959628894767772970, 6.45423508559015539058118507184, 7.19444929522728633540087632574, 7.85347257344309858946451212904

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