Properties

Label 2-8400-1.1-c1-0-45
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 − 6.31·11-s + 6.96·13-s + 6.57·17-s + 3.73·19-s + 21-s − 5.73·23-s + 27-s − 2·29-s + 1.03·31-s − 6.31·33-s + 10.7·37-s + 6.96·39-s − 6.96·41-s + 5.92·43-s + 49-s + 6.57·51-s + 1.03·53-s + 3.73·57-s + 3.22·59-s − 13.8·61-s + 63-s + 4.77·67-s − 5.73·69-s − 8.23·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 0.333·9-s − 1.90·11-s + 1.93·13-s + 1.59·17-s + 0.857·19-s + 0.218·21-s − 1.19·23-s + 0.192·27-s − 0.371·29-s + 0.186·31-s − 1.09·33-s + 1.75·37-s + 1.11·39-s − 1.08·41-s + 0.903·43-s + 0.142·49-s + 0.920·51-s + 0.142·53-s + 0.495·57-s + 0.419·59-s − 1.77·61-s + 0.125·63-s + 0.583·67-s − 0.690·69-s − 0.977·71-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

\(L(1)\) \(\approx\) \(2.906945235\)
\(L(\frac12)\) \(\approx\) \(2.906945235\)
\(L(\frac{3}{2})\) 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 \( 1 + 6.31T + 11T^{2} \)
13 \( 1 - 6.96T + 13T^{2} \)
17 \( 1 - 6.57T + 17T^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + 5.73T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 1.03T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 - 5.92T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 - 3.22T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 4.77T + 67T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 - 4.26T + 73T^{2} \)
79 \( 1 - 5.92T + 79T^{2} \)
83 \( 1 + 3.22T + 83T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 - 3.73T + 97T^{2} \)
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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

−7.76347363552739064434067078854, −7.56633736897097594093567312434, −6.24434360217085900286915605009, −5.72384291173585947344252326255, −5.10679529655927238229789347858, −4.13216278606118092736492517075, −3.38663535161985769080740079420, −2.78410955200357908504867536212, −1.76776781155872865809920997931, −0.843999041201389317655126401699, 0.843999041201389317655126401699, 1.76776781155872865809920997931, 2.78410955200357908504867536212, 3.38663535161985769080740079420, 4.13216278606118092736492517075, 5.10679529655927238229789347858, 5.72384291173585947344252326255, 6.24434360217085900286915605009, 7.56633736897097594093567312434, 7.76347363552739064434067078854

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