Properties

Label 2-4200-1.1-c1-0-40
Degree $2$
Conductor $4200$
Sign $-1$
Analytic cond. $33.5371$
Root an. cond. $5.79112$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 5.41·11-s − 4.34·13-s − 1.07·17-s + 4.34·19-s + 21-s − 6.34·23-s − 27-s − 8.83·29-s − 4.34·31-s − 5.41·33-s + 8.68·37-s + 4.34·39-s + 8.34·41-s − 6.15·43-s + 6.83·47-s + 49-s + 1.07·51-s − 6.18·53-s − 4.34·57-s − 6.83·59-s − 4.52·61-s − 63-s + 6.34·69-s − 14.0·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 0.333·9-s + 1.63·11-s − 1.20·13-s − 0.261·17-s + 0.995·19-s + 0.218·21-s − 1.32·23-s − 0.192·27-s − 1.64·29-s − 0.779·31-s − 0.943·33-s + 1.42·37-s + 0.694·39-s + 1.30·41-s − 0.938·43-s + 0.997·47-s + 0.142·49-s + 0.151·51-s − 0.849·53-s − 0.574·57-s − 0.890·59-s − 0.579·61-s − 0.125·63-s + 0.763·69-s − 1.67·71-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: no
Arithmetic: yes
Character: $\chi_{4200} (1, \cdot )$
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 \( 1 - 5.41T + 11T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
17 \( 1 + 1.07T + 17T^{2} \)
19 \( 1 - 4.34T + 19T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 + 8.83T + 29T^{2} \)
31 \( 1 + 4.34T + 31T^{2} \)
37 \( 1 - 8.68T + 37T^{2} \)
41 \( 1 - 8.34T + 41T^{2} \)
43 \( 1 + 6.15T + 43T^{2} \)
47 \( 1 - 6.83T + 47T^{2} \)
53 \( 1 + 6.18T + 53T^{2} \)
59 \( 1 + 6.83T + 59T^{2} \)
61 \( 1 + 4.52T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 - 11.1T + 73T^{2} \)
79 \( 1 - 0.680T + 79T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 - 6.49T + 89T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−7.69293959077972303293268548239, −7.42291578674046365785240599840, −6.41821501354579640339154606050, −5.97838545750475262426520244071, −5.08595788298688738686373603200, −4.19665787990758347454261913576, −3.59216739070791161437655696357, −2.36015385608100650414984528349, −1.33179347039825393918165419925, 0, 1.33179347039825393918165419925, 2.36015385608100650414984528349, 3.59216739070791161437655696357, 4.19665787990758347454261913576, 5.08595788298688738686373603200, 5.97838545750475262426520244071, 6.41821501354579640339154606050, 7.42291578674046365785240599840, 7.69293959077972303293268548239

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