Properties

Degree $2$
Conductor $4200$
Sign $1$
Motivic weight $1$
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 − 5·11-s + 4·13-s + 2·17-s + 4·19-s − 21-s + 9·23-s + 27-s − 5·29-s + 2·31-s − 5·33-s − 11·37-s + 4·39-s + 8·41-s − 11·43-s + 12·47-s + 49-s + 2·51-s − 10·53-s + 4·57-s + 12·59-s − 12·61-s − 63-s + 3·67-s + 9·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s + 0.485·17-s + 0.917·19-s − 0.218·21-s + 1.87·23-s + 0.192·27-s − 0.928·29-s + 0.359·31-s − 0.870·33-s − 1.80·37-s + 0.640·39-s + 1.24·41-s − 1.67·43-s + 1.75·47-s + 1/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.56·59-s − 1.53·61-s − 0.125·63-s + 0.366·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.278606686\)
\(L(\frac12)\) \(\approx\) \(2.278606686\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−18.32534145053861, −17.42199513811764, −16.76590693333899, −15.97298614355783, −15.62630464732029, −15.12688708438589, −14.22286727862321, −13.66388566630364, −13.06935383382471, −12.72604662269701, −11.76666433899194, −10.96255097863269, −10.47466636012317, −9.766381016180160, −8.968053365295563, −8.513027144263135, −7.557344946714562, −7.240556802397779, −6.165826073340002, −5.385212088043876, −4.753938716233858, −3.396909111141686, −3.206348842900314, −2.053241298536148, −0.8415597548309247, 0.8415597548309247, 2.053241298536148, 3.206348842900314, 3.396909111141686, 4.753938716233858, 5.385212088043876, 6.165826073340002, 7.240556802397779, 7.557344946714562, 8.513027144263135, 8.968053365295563, 9.766381016180160, 10.47466636012317, 10.96255097863269, 11.76666433899194, 12.72604662269701, 13.06935383382471, 13.66388566630364, 14.22286727862321, 15.12688708438589, 15.62630464732029, 15.97298614355783, 16.76590693333899, 17.42199513811764, 18.32534145053861

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