Properties

Label 4-140e2-1.1-c0e2-0-0
Degree $4$
Conductor $19600$
Sign $1$
Analytic cond. $0.00488169$
Root an. cond. $0.264327$
Motivic weight $0$
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 − 7-s + 8-s + 9-s + 10-s + 14-s − 15-s − 16-s − 18-s − 21-s + 23-s + 24-s + 2·27-s − 2·29-s + 30-s + 35-s − 40-s − 2·41-s + 42-s − 2·43-s − 45-s − 46-s − 2·47-s − 48-s + ⋯
L(s)  = 1  − 2-s + 3-s − 5-s − 6-s − 7-s + 8-s + 9-s + 10-s + 14-s − 15-s − 16-s − 18-s − 21-s + 23-s + 24-s + 2·27-s − 2·29-s + 30-s + 35-s − 40-s − 2·41-s + 42-s − 2·43-s − 45-s − 46-s − 2·47-s − 48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.00488169\)
Root analytic conductor: \(0.264327\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{140} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19600,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2644285454\)
\(L(\frac12)\) \(\approx\) \(0.2644285454\)
\(L(1)\) 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$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 + T + T^{2} \)
7$C_2$ \( 1 + T + T^{2} \)
good3$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
11$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
29$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + T^{2} )^{2} \)
47$C_2$ \( ( 1 + T + T^{2} )^{2} \)
53$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
67$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_2$ \( ( 1 + T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−14.17587321847294524728955915446, −13.02974905280989370564192259687, −12.96999685799762280518674606496, −12.17860024544497028843624594850, −11.42040436361054528192267196081, −11.16716525752643526132281201912, −10.27748392754102519390547451999, −9.877769206049269076046763875976, −9.614591293890291152644650722770, −8.868601555853835613352467318640, −8.416143973274795439126093561837, −8.180747780401275503283824113235, −7.26104126294525014046992686963, −7.10387591591902298203109404153, −6.41661967871024913557960236053, −5.13376199973804018896584366820, −4.57530209214505313204504732398, −3.43782950479146480647989201129, −3.41763793665203501307430262132, −1.84798733176887999444039248112, 1.84798733176887999444039248112, 3.41763793665203501307430262132, 3.43782950479146480647989201129, 4.57530209214505313204504732398, 5.13376199973804018896584366820, 6.41661967871024913557960236053, 7.10387591591902298203109404153, 7.26104126294525014046992686963, 8.180747780401275503283824113235, 8.416143973274795439126093561837, 8.868601555853835613352467318640, 9.614591293890291152644650722770, 9.877769206049269076046763875976, 10.27748392754102519390547451999, 11.16716525752643526132281201912, 11.42040436361054528192267196081, 12.17860024544497028843624594850, 12.96999685799762280518674606496, 13.02974905280989370564192259687, 14.17587321847294524728955915446

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