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Properties

Label 2-39600-1.1-c1-0-132
Degree $2$
Conductor $39600$
Sign $-1$
Analytic cond. $316.207$
Root an. cond. $17.7822$
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  + 11-s + 4·13-s + 8·19-s + 4·23-s − 2·29-s − 4·31-s + 8·37-s + 2·41-s − 4·47-s − 7·49-s − 12·53-s − 12·59-s + 2·61-s − 12·67-s − 12·73-s − 12·79-s − 8·83-s − 6·89-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.301·11-s + 1.10·13-s + 1.83·19-s + 0.834·23-s − 0.371·29-s − 0.718·31-s + 1.31·37-s + 0.312·41-s − 0.583·47-s − 49-s − 1.64·53-s − 1.56·59-s + 0.256·61-s − 1.46·67-s − 1.40·73-s − 1.35·79-s − 0.878·83-s − 0.635·89-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(316.207\)
Root analytic conductor: \(17.7822\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 39600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
11 \( 1 - T \)
good7 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−15.04717200891448, −14.42469740229480, −14.10852599434828, −13.45911925198224, −12.98737757224037, −12.60710469221000, −11.76174845637193, −11.28578906385935, −11.14424282194741, −10.27056656802770, −9.744360113252109, −9.125502419205054, −8.887772885874551, −7.974554222942420, −7.591858278192601, −7.042683133889821, −6.201482027052304, −5.935034198601166, −5.141529120819849, −4.601647210239732, −3.830969194139639, −3.199004532995094, −2.770810019376989, −1.433030391958796, −1.294571315249983, 0, 1.294571315249983, 1.433030391958796, 2.770810019376989, 3.199004532995094, 3.830969194139639, 4.601647210239732, 5.141529120819849, 5.935034198601166, 6.201482027052304, 7.042683133889821, 7.591858278192601, 7.974554222942420, 8.887772885874551, 9.125502419205054, 9.744360113252109, 10.27056656802770, 11.14424282194741, 11.28578906385935, 11.76174845637193, 12.60710469221000, 12.98737757224037, 13.45911925198224, 14.10852599434828, 14.42469740229480, 15.04717200891448

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