Properties

Label 4-3450e2-1.1-c1e2-0-2
Degree $4$
Conductor $11902500$
Sign $1$
Analytic cond. $758.913$
Root an. cond. $5.24865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s − 4·11-s + 16-s + 8·29-s + 36-s + 12·41-s + 4·44-s − 2·49-s − 24·59-s − 28·61-s − 64-s − 4·71-s − 16·79-s + 81-s + 16·89-s + 4·99-s − 24·101-s − 28·109-s − 8·116-s − 10·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 1.48·29-s + 1/6·36-s + 1.87·41-s + 0.603·44-s − 2/7·49-s − 3.12·59-s − 3.58·61-s − 1/8·64-s − 0.474·71-s − 1.80·79-s + 1/9·81-s + 1.69·89-s + 0.402·99-s − 2.38·101-s − 2.68·109-s − 0.742·116-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11902500\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(758.913\)
Root analytic conductor: \(5.24865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3450} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4660047342\)
\(L(\frac12)\) \(\approx\) \(0.4660047342\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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

−9.048310987488586818230960361768, −8.369749158830849622726163695797, −7.889023824608979117311818396722, −7.76860115865797413948480623479, −7.54491682378360765865229958244, −6.94609393656072801726459869821, −6.36392212570722653978851548852, −6.19594758454851536533325490384, −5.80110155840610961592399105441, −5.33060200352736723320980304836, −4.95330297041993171173896061820, −4.49330831004622263226039870989, −4.36831178494353495923880787683, −3.72083375007140684079021589082, −3.09173744627403608460008293702, −2.69721655858785123884089056511, −2.64156845921894809913343354082, −1.57849636336488535884124184564, −1.24595322760943312660261276553, −0.21512056250361237315298819939, 0.21512056250361237315298819939, 1.24595322760943312660261276553, 1.57849636336488535884124184564, 2.64156845921894809913343354082, 2.69721655858785123884089056511, 3.09173744627403608460008293702, 3.72083375007140684079021589082, 4.36831178494353495923880787683, 4.49330831004622263226039870989, 4.95330297041993171173896061820, 5.33060200352736723320980304836, 5.80110155840610961592399105441, 6.19594758454851536533325490384, 6.36392212570722653978851548852, 6.94609393656072801726459869821, 7.54491682378360765865229958244, 7.76860115865797413948480623479, 7.889023824608979117311818396722, 8.369749158830849622726163695797, 9.048310987488586818230960361768

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