Properties

Label 4-3840e2-1.1-c1e2-0-56
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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 + 2·5-s + 3·9-s + 4·13-s + 4·15-s − 25-s + 4·27-s + 16·31-s − 4·37-s + 8·39-s + 12·41-s − 8·43-s + 6·45-s + 10·49-s + 12·53-s + 8·65-s − 8·71-s − 2·75-s − 16·79-s + 5·81-s + 24·83-s + 28·89-s + 32·93-s + 24·107-s − 8·111-s + 12·117-s − 14·121-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 9-s + 1.10·13-s + 1.03·15-s − 1/5·25-s + 0.769·27-s + 2.87·31-s − 0.657·37-s + 1.28·39-s + 1.87·41-s − 1.21·43-s + 0.894·45-s + 10/7·49-s + 1.64·53-s + 0.992·65-s − 0.949·71-s − 0.230·75-s − 1.80·79-s + 5/9·81-s + 2.63·83-s + 2.96·89-s + 3.31·93-s + 2.32·107-s − 0.759·111-s + 1.10·117-s − 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.418833751\)
\(L(\frac12)\) \(\approx\) \(7.418833751\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 2 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.733820273216914718701308146871, −8.382924299285156204049586835220, −7.920632073914627816261596715370, −7.77353422828751937920084895569, −7.21551074569229676937747824172, −6.88999540728944346452539273545, −6.33824499941308818949989329903, −6.17459060460245651449405104106, −5.84875309917937703041865679390, −5.28696956116940240275729054352, −4.86428778680526759388612534048, −4.39189430287182709092506074615, −4.03516442239569406157107829835, −3.62159602338295760947142595201, −3.09016986958976323889115787108, −2.76891566852514825993641777777, −2.12576054785698034647156900768, −2.02482757013089088582550482867, −1.10440756254888688392528701383, −0.854119067424763064850588944514, 0.854119067424763064850588944514, 1.10440756254888688392528701383, 2.02482757013089088582550482867, 2.12576054785698034647156900768, 2.76891566852514825993641777777, 3.09016986958976323889115787108, 3.62159602338295760947142595201, 4.03516442239569406157107829835, 4.39189430287182709092506074615, 4.86428778680526759388612534048, 5.28696956116940240275729054352, 5.84875309917937703041865679390, 6.17459060460245651449405104106, 6.33824499941308818949989329903, 6.88999540728944346452539273545, 7.21551074569229676937747824172, 7.77353422828751937920084895569, 7.920632073914627816261596715370, 8.382924299285156204049586835220, 8.733820273216914718701308146871

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