Properties

Label 4-4800e2-1.1-c1e2-0-34
Degree $4$
Conductor $23040000$
Sign $1$
Analytic cond. $1469.05$
Root an. cond. $6.19097$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 4·11-s + 10·19-s + 20·29-s + 6·31-s − 16·41-s + 5·49-s + 20·59-s − 14·61-s + 16·71-s + 81-s − 4·99-s − 24·101-s + 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s − 10·171-s + 173-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.20·11-s + 2.29·19-s + 3.71·29-s + 1.07·31-s − 2.49·41-s + 5/7·49-s + 2.60·59-s − 1.79·61-s + 1.89·71-s + 1/9·81-s − 0.402·99-s − 2.38·101-s + 0.957·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s − 0.764·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23040000\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1469.05\)
Root analytic conductor: \(6.19097\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.934236567\)
\(L(\frac12)\) \(\approx\) \(3.934236567\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 95 T^{2} + p^{2} T^{4} \)
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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

−8.589124827348481471833685488340, −8.048787373294178350620862535016, −8.003492524450074869667556481498, −7.17308620956541651447560497629, −6.95669667609099563108475656357, −6.77931766684523473609719553630, −6.28882914090260720125246913109, −6.06771731384028623477623885545, −5.45887540204000471691637356250, −4.98901427664835785807480654895, −4.98652134563867613813000472289, −4.38681589508485589073896269040, −3.94990396447826469898531124589, −3.38041314016615848364295095934, −3.23311766744514547543812781387, −2.64447477526328708860233085579, −2.34344732904433365577925785050, −1.31374435277710617947574216176, −1.22143241512701422392711250804, −0.62701116250275862884048503316, 0.62701116250275862884048503316, 1.22143241512701422392711250804, 1.31374435277710617947574216176, 2.34344732904433365577925785050, 2.64447477526328708860233085579, 3.23311766744514547543812781387, 3.38041314016615848364295095934, 3.94990396447826469898531124589, 4.38681589508485589073896269040, 4.98652134563867613813000472289, 4.98901427664835785807480654895, 5.45887540204000471691637356250, 6.06771731384028623477623885545, 6.28882914090260720125246913109, 6.77931766684523473609719553630, 6.95669667609099563108475656357, 7.17308620956541651447560497629, 8.003492524450074869667556481498, 8.048787373294178350620862535016, 8.589124827348481471833685488340

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