Properties

Label 4-140e2-1.1-c1e2-0-7
Degree $4$
Conductor $19600$
Sign $1$
Analytic cond. $1.24971$
Root an. cond. $1.05731$
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  + 3·3-s + 5-s + 7-s + 3·9-s + 2·11-s − 12·13-s + 3·15-s − 2·17-s + 3·21-s + 9·23-s + 6·29-s − 2·31-s + 6·33-s + 35-s − 8·37-s − 36·39-s + 10·41-s + 2·43-s + 3·45-s − 8·47-s − 6·49-s − 6·51-s − 4·53-s + 2·55-s + 8·59-s − 7·61-s + 3·63-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 0.377·7-s + 9-s + 0.603·11-s − 3.32·13-s + 0.774·15-s − 0.485·17-s + 0.654·21-s + 1.87·23-s + 1.11·29-s − 0.359·31-s + 1.04·33-s + 0.169·35-s − 1.31·37-s − 5.76·39-s + 1.56·41-s + 0.304·43-s + 0.447·45-s − 1.16·47-s − 6/7·49-s − 0.840·51-s − 0.549·53-s + 0.269·55-s + 1.04·59-s − 0.896·61-s + 0.377·63-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(19600\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.24971\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{140} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.927964035\)
\(L(\frac12)\) \(\approx\) \(1.927964035\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - T + p T^{2} \)
good3$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−13.60122216544441798159624896179, −12.94793081897094035412996021100, −12.34319715475179651851535376233, −12.26705420246019802564831307460, −11.32006546482095040395029376052, −10.89001663872321024420385947457, −10.02227595596695412511737405192, −9.567992273346978297347743560120, −9.406135851580823636168006377522, −8.696057970109697825872006184581, −8.299185265549282275363671131429, −7.61817640236334635123069039669, −7.04975275008661300422987347886, −6.76744508605909680024719185851, −5.49297177427976641054165719345, −4.86887653197319161164792302111, −4.38522254492250913918981177806, −3.06969582227842589892302745920, −2.73897840237183253770951264479, −1.95938077675247174214824466243, 1.95938077675247174214824466243, 2.73897840237183253770951264479, 3.06969582227842589892302745920, 4.38522254492250913918981177806, 4.86887653197319161164792302111, 5.49297177427976641054165719345, 6.76744508605909680024719185851, 7.04975275008661300422987347886, 7.61817640236334635123069039669, 8.299185265549282275363671131429, 8.696057970109697825872006184581, 9.406135851580823636168006377522, 9.567992273346978297347743560120, 10.02227595596695412511737405192, 10.89001663872321024420385947457, 11.32006546482095040395029376052, 12.26705420246019802564831307460, 12.34319715475179651851535376233, 12.94793081897094035412996021100, 13.60122216544441798159624896179

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