Properties

Label 4-3360e2-1.1-c1e2-0-15
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4·5-s + 3·9-s + 12·13-s + 8·15-s + 11·25-s + 4·27-s + 16·31-s + 4·37-s + 24·39-s − 16·41-s − 12·43-s + 12·45-s − 49-s + 12·53-s + 48·65-s − 4·67-s + 16·71-s + 22·75-s − 20·79-s + 5·81-s + 8·83-s + 32·93-s − 24·107-s + 8·111-s + 36·117-s + 22·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 9-s + 3.32·13-s + 2.06·15-s + 11/5·25-s + 0.769·27-s + 2.87·31-s + 0.657·37-s + 3.84·39-s − 2.49·41-s − 1.82·43-s + 1.78·45-s − 1/7·49-s + 1.64·53-s + 5.95·65-s − 0.488·67-s + 1.89·71-s + 2.54·75-s − 2.25·79-s + 5/9·81-s + 0.878·83-s + 3.31·93-s − 2.32·107-s + 0.759·111-s + 3.32·117-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.975133900\)
\(L(\frac12)\) \(\approx\) \(9.975133900\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.581560827924849228781922146356, −8.441411168293010299692553632695, −8.222697929943080246071652705327, −8.182357158335574345907535022590, −7.01742851937069154793296378618, −7.00916532230518471348308326378, −6.37530601438608607827372528320, −6.35979175866179464745774511205, −5.85410286136060566975694943057, −5.57579470116411348658055023530, −4.80713933950394354196722973975, −4.76306966282468476764715853463, −3.86079552578928982012256439025, −3.75845353347607572359956385098, −3.07809411424618637920148144050, −2.93719853611986329628123122159, −2.22080543800085739800945322341, −1.79951009396895517506081376126, −1.14962749118117861696458884888, −1.12377073257572614649069469309, 1.12377073257572614649069469309, 1.14962749118117861696458884888, 1.79951009396895517506081376126, 2.22080543800085739800945322341, 2.93719853611986329628123122159, 3.07809411424618637920148144050, 3.75845353347607572359956385098, 3.86079552578928982012256439025, 4.76306966282468476764715853463, 4.80713933950394354196722973975, 5.57579470116411348658055023530, 5.85410286136060566975694943057, 6.35979175866179464745774511205, 6.37530601438608607827372528320, 7.00916532230518471348308326378, 7.01742851937069154793296378618, 8.182357158335574345907535022590, 8.222697929943080246071652705327, 8.441411168293010299692553632695, 8.581560827924849228781922146356

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