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Properties

Label 2-6300-1.1-c1-0-45
Degree $2$
Conductor $6300$
Sign $-1$
Analytic cond. $50.3057$
Root an. cond. $7.09265$
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  + 7-s + 4·13-s − 4·17-s + 4·19-s − 8·23-s − 2·29-s − 8·31-s − 8·37-s − 6·41-s + 8·43-s − 8·47-s + 49-s + 4·59-s − 6·61-s + 8·67-s − 12·71-s − 4·73-s − 4·79-s + 10·89-s + 4·91-s − 12·97-s + 18·101-s − 8·103-s − 8·107-s + 14·109-s + 16·113-s − 4·119-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s − 0.970·17-s + 0.917·19-s − 1.66·23-s − 0.371·29-s − 1.43·31-s − 1.31·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 1/7·49-s + 0.520·59-s − 0.768·61-s + 0.977·67-s − 1.42·71-s − 0.468·73-s − 0.450·79-s + 1.05·89-s + 0.419·91-s − 1.21·97-s + 1.79·101-s − 0.788·103-s − 0.773·107-s + 1.34·109-s + 1.50·113-s − 0.366·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(50.3057\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 6300,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−7.67054901454349574389188410869, −7.05205376829404405112199417011, −6.18741467938917119399756921892, −5.62735129748220093625365788805, −4.81296937316620313566688024034, −3.92557973255562052824925707090, −3.37908019863937010502830023430, −2.14454538267914275288942031565, −1.44834759856195077658271204870, 0, 1.44834759856195077658271204870, 2.14454538267914275288942031565, 3.37908019863937010502830023430, 3.92557973255562052824925707090, 4.81296937316620313566688024034, 5.62735129748220093625365788805, 6.18741467938917119399756921892, 7.05205376829404405112199417011, 7.67054901454349574389188410869

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