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Properties

Label 2-23400-1.1-c1-0-36
Degree $2$
Conductor $23400$
Sign $-1$
Analytic cond. $186.849$
Root an. cond. $13.6693$
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  + 13-s + 2·17-s + 4·23-s − 6·29-s + 8·31-s − 10·37-s + 6·41-s − 4·43-s − 8·47-s − 7·49-s + 2·53-s + 6·61-s − 12·67-s − 8·71-s − 6·73-s + 8·79-s − 12·83-s + 14·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.277·13-s + 0.485·17-s + 0.834·23-s − 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 49-s + 0.274·53-s + 0.768·61-s − 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.900·79-s − 1.31·83-s + 1.48·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(23400\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(186.849\)
Root analytic conductor: \(13.6693\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 23400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−15.84546222444048, −15.08029332174249, −14.71180703247948, −14.16241432164870, −13.46141995399464, −13.11555575419780, −12.53375812072280, −11.77120386793558, −11.54436005043722, −10.73577698579194, −10.29306253414945, −9.698921272888221, −9.052430320694950, −8.564580022424441, −7.894281434353351, −7.358720123375471, −6.666402526526724, −6.149791376880309, −5.373273924828569, −4.891118516298287, −4.113093752562562, −3.369702548577172, −2.838807527997471, −1.835700215904297, −1.135438720991837, 0, 1.135438720991837, 1.835700215904297, 2.838807527997471, 3.369702548577172, 4.113093752562562, 4.891118516298287, 5.373273924828569, 6.149791376880309, 6.666402526526724, 7.358720123375471, 7.894281434353351, 8.564580022424441, 9.052430320694950, 9.698921272888221, 10.29306253414945, 10.73577698579194, 11.54436005043722, 11.77120386793558, 12.53375812072280, 13.11555575419780, 13.46141995399464, 14.16241432164870, 14.71180703247948, 15.08029332174249, 15.84546222444048

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