Properties

Label 4-570e2-1.1-c1e2-0-10
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 + 6·7-s + 8-s − 10-s + 2·11-s − 2·13-s − 6·14-s + 15-s − 16-s − 2·17-s + 19-s + 6·21-s − 2·22-s − 23-s + 24-s + 2·26-s − 27-s − 30-s + 8·31-s + 2·33-s + 2·34-s + 6·35-s + 18·37-s − 38-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.447·5-s − 0.408·6-s + 2.26·7-s + 0.353·8-s − 0.316·10-s + 0.603·11-s − 0.554·13-s − 1.60·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s + 0.229·19-s + 1.30·21-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.182·30-s + 1.43·31-s + 0.348·33-s + 0.342·34-s + 1.01·35-s + 2.95·37-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

\(L(1)\) \(\approx\) \(2.250988730\)
\(L(\frac12)\) \(\approx\) \(2.250988730\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 5 T - 64 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
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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

−10.86216630797830197771544462783, −10.63426663744011888226285355104, −9.862702111902970869832230328471, −9.539265900986971273021900383104, −9.283714697770783243830820782476, −8.706002306678804249323668443411, −8.211573016864783301744359962197, −8.050848118267645269781345689289, −7.58387040399957933189939885123, −7.26230196400345856843339444675, −6.23133151202982325455788087958, −6.22577779149992644484561152965, −5.21451842914696800654131433262, −4.91159260837037707326470607195, −4.27949388648767022068077991604, −4.08675263013582426957232220789, −2.79016266190462774502875364236, −2.42690397155486140022543270399, −1.61826355711904235052075181476, −1.07104913896236154595263976823, 1.07104913896236154595263976823, 1.61826355711904235052075181476, 2.42690397155486140022543270399, 2.79016266190462774502875364236, 4.08675263013582426957232220789, 4.27949388648767022068077991604, 4.91159260837037707326470607195, 5.21451842914696800654131433262, 6.22577779149992644484561152965, 6.23133151202982325455788087958, 7.26230196400345856843339444675, 7.58387040399957933189939885123, 8.050848118267645269781345689289, 8.211573016864783301744359962197, 8.706002306678804249323668443411, 9.283714697770783243830820782476, 9.539265900986971273021900383104, 9.862702111902970869832230328471, 10.63426663744011888226285355104, 10.86216630797830197771544462783

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