Properties

Label 4-570e2-1.1-c1e2-0-0
Degree $4$
Conductor $324900$
Sign $1$
Analytic cond. $20.7159$
Root an. cond. $2.13341$
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  + 2-s + 3-s − 5-s + 6-s − 10·7-s − 8-s − 10-s + 2·11-s − 6·13-s − 10·14-s − 15-s − 16-s − 4·17-s − 19-s − 10·21-s + 2·22-s − 7·23-s − 24-s − 6·26-s − 27-s − 6·29-s − 30-s + 2·33-s − 4·34-s + 10·35-s + 14·37-s − 38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 3.77·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s − 2.67·14-s − 0.258·15-s − 1/4·16-s − 0.970·17-s − 0.229·19-s − 2.18·21-s + 0.426·22-s − 1.45·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s − 1.11·29-s − 0.182·30-s + 0.348·33-s − 0.685·34-s + 1.69·35-s + 2.30·37-s − 0.162·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(324900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(20.7159\)
Root analytic conductor: \(2.13341\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{570} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 324900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3069413378\)
\(L(\frac12)\) \(\approx\) \(0.3069413378\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good7$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 2 T - 67 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−11.42869921966167518258910561516, −10.04197239434909792433143160226, −9.974155174672416300948973208773, −9.802336231105367834325175494434, −9.257169270914479051718540195140, −8.974280613717650388146540694003, −8.492503754776798021061565107724, −7.53415357675152026962172304638, −7.34614225563783647250492057108, −6.84407819521152970351372081234, −6.35151653539977994956861521099, −5.92652352799652711155296294435, −5.77800025558536022715943352408, −4.59545782583814197203488182503, −4.14101402507608497519415351861, −3.77351532917281946157175500922, −3.23698953885078968562930083930, −2.65775782371746890693081722965, −2.38477296262791958911702648945, −0.25043747681281072754802364254, 0.25043747681281072754802364254, 2.38477296262791958911702648945, 2.65775782371746890693081722965, 3.23698953885078968562930083930, 3.77351532917281946157175500922, 4.14101402507608497519415351861, 4.59545782583814197203488182503, 5.77800025558536022715943352408, 5.92652352799652711155296294435, 6.35151653539977994956861521099, 6.84407819521152970351372081234, 7.34614225563783647250492057108, 7.53415357675152026962172304638, 8.492503754776798021061565107724, 8.974280613717650388146540694003, 9.257169270914479051718540195140, 9.802336231105367834325175494434, 9.974155174672416300948973208773, 10.04197239434909792433143160226, 11.42869921966167518258910561516

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