Properties

Label 2-8400-1.1-c1-0-6
Degree $2$
Conductor $8400$
Sign $1$
Analytic cond. $67.0743$
Root an. cond. $8.18989$
Motivic weight $1$
Arithmetic yes
Rational yes
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 − 4·11-s − 2·13-s − 2·17-s + 2·19-s − 21-s − 6·23-s − 27-s + 6·29-s − 6·31-s + 4·33-s + 4·37-s + 2·39-s − 4·43-s + 4·47-s + 49-s + 2·51-s + 2·53-s − 2·57-s − 4·59-s − 2·61-s + 63-s − 12·67-s + 6·69-s + 8·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 1.11·29-s − 1.07·31-s + 0.696·33-s + 0.657·37-s + 0.320·39-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.264·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 1.46·67-s + 0.722·69-s + 0.949·71-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050898657\)
\(L(\frac12)\) \(\approx\) \(1.050898657\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−7.56279277731945914272912400131, −7.35048406859826782523602071262, −6.23306806472184064493666247970, −5.78489214933156331783686829675, −4.89697965516847316969092004032, −4.57114548767155620108541318156, −3.50101908861801476802011770116, −2.55181171513341671521163672538, −1.79768067173493383952295287520, −0.50273118136580517687805037565, 0.50273118136580517687805037565, 1.79768067173493383952295287520, 2.55181171513341671521163672538, 3.50101908861801476802011770116, 4.57114548767155620108541318156, 4.89697965516847316969092004032, 5.78489214933156331783686829675, 6.23306806472184064493666247970, 7.35048406859826782523602071262, 7.56279277731945914272912400131

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