Properties

Label 4-60e4-1.1-c1e2-0-22
Degree $4$
Conductor $12960000$
Sign $1$
Analytic cond. $826.340$
Root an. cond. $5.36154$
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  + 16·19-s + 8·31-s − 2·49-s + 28·61-s − 8·79-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯
L(s)  = 1  + 3.67·19-s + 1.43·31-s − 2/7·49-s + 3.58·61-s − 0.900·79-s − 0.383·109-s − 2·121-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 + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12960000\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(826.340\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.539006381\)
\(L(\frac12)\) \(\approx\) \(3.539006381\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2$ \( ( 1 - p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
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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

−8.679854463838395859295167662628, −8.302179430291873342597403143302, −8.016459470359899075972712848442, −7.57586021832068742494217947493, −7.28101304657493281489624231555, −7.03241617073278871019508592207, −6.36598383165525711690532783933, −6.34501718621439964387021312327, −5.46019200520156091419249562695, −5.27436891853782060412211726730, −5.27390804844714373075996193123, −4.57373705258790200882689187094, −3.99678718250803551140047687598, −3.74246408996166194370495845381, −3.02275634312633459334619558189, −2.97711038721379082663734096602, −2.38245825924528526673071130489, −1.61928431382313917897956963915, −1.06556667006186634666569473452, −0.67429971704453545005553088813, 0.67429971704453545005553088813, 1.06556667006186634666569473452, 1.61928431382313917897956963915, 2.38245825924528526673071130489, 2.97711038721379082663734096602, 3.02275634312633459334619558189, 3.74246408996166194370495845381, 3.99678718250803551140047687598, 4.57373705258790200882689187094, 5.27390804844714373075996193123, 5.27436891853782060412211726730, 5.46019200520156091419249562695, 6.34501718621439964387021312327, 6.36598383165525711690532783933, 7.03241617073278871019508592207, 7.28101304657493281489624231555, 7.57586021832068742494217947493, 8.016459470359899075972712848442, 8.302179430291873342597403143302, 8.679854463838395859295167662628

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