Properties

Label 2-4200-1.1-c1-0-35
Degree $2$
Conductor $4200$
Sign $-1$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 7-s + 9-s − 3·11-s − 2·13-s + 4·17-s − 2·19-s + 21-s + 9·23-s − 27-s − 7·29-s + 10·31-s + 3·33-s − 5·37-s + 2·39-s − 8·41-s + 7·43-s + 10·47-s + 49-s − 4·51-s − 10·53-s + 2·57-s − 6·59-s − 8·61-s − 63-s + 13·67-s − 9·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.970·17-s − 0.458·19-s + 0.218·21-s + 1.87·23-s − 0.192·27-s − 1.29·29-s + 1.79·31-s + 0.522·33-s − 0.821·37-s + 0.320·39-s − 1.24·41-s + 1.06·43-s + 1.45·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.264·57-s − 0.781·59-s − 1.02·61-s − 0.125·63-s + 1.58·67-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: $-1$
Analytic conductor: \(33.5371\)
Root analytic conductor: \(5.79112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 13 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−7.899473909663191807572903112462, −7.27720131383344427771370987871, −6.61351801028347888938198222606, −5.70297484775777887400292195384, −5.16017802130727569371052854335, −4.40891702746943177258309030421, −3.28568334008219714322200718461, −2.57096142035794862492835757112, −1.24547113208558814577415151922, 0, 1.24547113208558814577415151922, 2.57096142035794862492835757112, 3.28568334008219714322200718461, 4.40891702746943177258309030421, 5.16017802130727569371052854335, 5.70297484775777887400292195384, 6.61351801028347888938198222606, 7.27720131383344427771370987871, 7.899473909663191807572903112462

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