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Properties

Label 2-117600-1.1-c1-0-106
Degree $2$
Conductor $117600$
Sign $-1$
Analytic cond. $939.040$
Root an. cond. $30.6437$
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  − 3-s + 9-s − 2·11-s − 2·13-s + 6·19-s − 8·23-s − 27-s + 6·29-s − 10·31-s + 2·33-s + 8·37-s + 2·39-s + 6·41-s + 4·43-s + 8·47-s − 2·53-s − 6·57-s + 12·59-s − 14·61-s + 8·67-s + 8·69-s + 2·71-s − 2·73-s − 8·79-s + 81-s − 6·87-s − 6·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 1.37·19-s − 1.66·23-s − 0.192·27-s + 1.11·29-s − 1.79·31-s + 0.348·33-s + 1.31·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s − 0.794·57-s + 1.56·59-s − 1.79·61-s + 0.977·67-s + 0.963·69-s + 0.237·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 0.643·87-s − 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117600\)    =    \(2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(939.040\)
Root analytic conductor: \(30.6437\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 117600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
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 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−13.84421485068230, −13.31631193796872, −12.72635533815678, −12.38701535371906, −11.89106670709133, −11.47988637007122, −10.88468106778963, −10.49081822330865, −9.930025556084186, −9.502187953358697, −9.113498078787123, −8.238029941138204, −7.835880016126346, −7.387979581967730, −6.942570776491956, −6.181409307667376, −5.655111903894351, −5.434002120687835, −4.671158310288742, −4.168317869999056, −3.618067134531663, −2.742897466060885, −2.372139039456345, −1.502820174571975, −0.7746284733450846, 0, 0.7746284733450846, 1.502820174571975, 2.372139039456345, 2.742897466060885, 3.618067134531663, 4.168317869999056, 4.671158310288742, 5.434002120687835, 5.655111903894351, 6.181409307667376, 6.942570776491956, 7.387979581967730, 7.835880016126346, 8.238029941138204, 9.113498078787123, 9.502187953358697, 9.930025556084186, 10.49081822330865, 10.88468106778963, 11.47988637007122, 11.89106670709133, 12.38701535371906, 12.72635533815678, 13.31631193796872, 13.84421485068230

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