Properties

Label 4-1950e2-1.1-c1e2-0-2
Degree $4$
Conductor $3802500$
Sign $1$
Analytic cond. $242.450$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 10·11-s + 16-s + 14·29-s − 18·31-s + 36-s − 4·41-s + 10·44-s + 13·49-s − 2·59-s − 14·61-s − 64-s − 16·71-s + 8·79-s + 81-s − 32·89-s + 10·99-s − 14·101-s − 12·109-s − 14·116-s + 53·121-s + 18·124-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 3.01·11-s + 1/4·16-s + 2.59·29-s − 3.23·31-s + 1/6·36-s − 0.624·41-s + 1.50·44-s + 13/7·49-s − 0.260·59-s − 1.79·61-s − 1/8·64-s − 1.89·71-s + 0.900·79-s + 1/9·81-s − 3.39·89-s + 1.00·99-s − 1.39·101-s − 1.14·109-s − 1.29·116-s + 4.81·121-s + 1.61·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3802500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(242.450\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3802500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1586405884\)
\(L(\frac12)\) \(\approx\) \(0.1586405884\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 91 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.358836541444472762023956318257, −8.880145600659618493736890585642, −8.560268597684493583990931931554, −8.268475352722482195552890199545, −7.81435049446838867369636564597, −7.51944799593593933739739001118, −7.16495357073504651045490759837, −6.72851077647448023587886233824, −5.99122045962487592630226658800, −5.69019976214886025208218714122, −5.25164082612100078630129655200, −5.16985933931257427817887475425, −4.48536724697619186947534976668, −4.18747228020501973519725178121, −3.41260050546260402018528298376, −2.91802923056800729184531683581, −2.68248858013743886723859883368, −2.07935729728428045054184120617, −1.27164726816683742350119633728, −0.14732165815762724911452578913, 0.14732165815762724911452578913, 1.27164726816683742350119633728, 2.07935729728428045054184120617, 2.68248858013743886723859883368, 2.91802923056800729184531683581, 3.41260050546260402018528298376, 4.18747228020501973519725178121, 4.48536724697619186947534976668, 5.16985933931257427817887475425, 5.25164082612100078630129655200, 5.69019976214886025208218714122, 5.99122045962487592630226658800, 6.72851077647448023587886233824, 7.16495357073504651045490759837, 7.51944799593593933739739001118, 7.81435049446838867369636564597, 8.268475352722482195552890199545, 8.560268597684493583990931931554, 8.880145600659618493736890585642, 9.358836541444472762023956318257

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