Properties

Label 4-450e2-1.1-c5e2-0-16
Degree $4$
Conductor $202500$
Sign $1$
Analytic cond. $5208.90$
Root an. cond. $8.49545$
Motivic weight $5$
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  − 16·4-s + 996·11-s + 256·16-s + 3.41e3·19-s + 1.56e4·29-s + 1.95e3·31-s + 1.62e4·41-s − 1.59e4·44-s − 2.06e4·49-s + 7.18e4·59-s + 7.05e3·61-s − 4.09e3·64-s + 1.50e4·71-s − 5.45e4·76-s + 4.54e4·79-s − 1.57e5·89-s − 2.11e5·101-s − 3.39e5·109-s − 2.50e5·116-s + 4.21e5·121-s − 3.12e4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1/2·4-s + 2.48·11-s + 1/4·16-s + 2.16·19-s + 3.45·29-s + 0.365·31-s + 1.51·41-s − 1.24·44-s − 1.23·49-s + 2.68·59-s + 0.242·61-s − 1/8·64-s + 0.355·71-s − 1.08·76-s + 0.819·79-s − 2.11·89-s − 2.05·101-s − 2.73·109-s − 1.72·116-s + 2.61·121-s − 0.182·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(202500\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5208.90\)
Root analytic conductor: \(8.49545\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 202500,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(5.219026312\)
\(L(\frac12)\) \(\approx\) \(5.219026312\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 20675 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 498 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 88105 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 1835710 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1705 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10457770 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 - 270 p T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 977 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 115436230 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 8148 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 + 85025075 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 387827290 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 - 524374090 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 - 35910 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 3527 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 602895515 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 7548 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 4145725870 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 - 22720 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7744123810 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 + 78960 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 14194284865 T^{2} + p^{10} T^{4} \)
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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

−10.39327360418251247698837314349, −9.894035240479646785678045928823, −9.481658581628181040781009738857, −9.394012638392848689329039917442, −8.707883497582276091591723809566, −8.266681119550813182480584871326, −7.982534451473953299366396006029, −7.10403391905505918909641437101, −6.74461202560975420272404410661, −6.51142085319768684907579093637, −5.78559822479078762602279715531, −5.28204674117611898410969703455, −4.67852629915443502086391780684, −4.16231082955827487946164769186, −3.77961938066780321635110940594, −3.05843028153110295655057029766, −2.60053570576630696783709110047, −1.42312071288491152235082666376, −1.05999910540025849415515964428, −0.70177569216353709191054009788, 0.70177569216353709191054009788, 1.05999910540025849415515964428, 1.42312071288491152235082666376, 2.60053570576630696783709110047, 3.05843028153110295655057029766, 3.77961938066780321635110940594, 4.16231082955827487946164769186, 4.67852629915443502086391780684, 5.28204674117611898410969703455, 5.78559822479078762602279715531, 6.51142085319768684907579093637, 6.74461202560975420272404410661, 7.10403391905505918909641437101, 7.982534451473953299366396006029, 8.266681119550813182480584871326, 8.707883497582276091591723809566, 9.394012638392848689329039917442, 9.481658581628181040781009738857, 9.894035240479646785678045928823, 10.39327360418251247698837314349

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