Properties

Label 4-3450e2-1.1-c1e2-0-19
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

Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 9-s + 4·11-s + 16-s + 8·19-s − 16·29-s + 16·31-s + 36-s + 12·41-s − 4·44-s − 2·49-s − 8·59-s + 12·61-s − 64-s + 20·71-s − 8·76-s + 16·79-s + 81-s − 4·99-s − 16·101-s + 28·109-s + 16·116-s − 10·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 1.83·19-s − 2.97·29-s + 2.87·31-s + 1/6·36-s + 1.87·41-s − 0.603·44-s − 2/7·49-s − 1.04·59-s + 1.53·61-s − 1/8·64-s + 2.37·71-s − 0.917·76-s + 1.80·79-s + 1/9·81-s − 0.402·99-s − 1.59·101-s + 2.68·109-s + 1.48·116-s − 0.909·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.095456373\)
\(L(\frac12)\) \(\approx\) \(3.095456373\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−8.911949685783390555619457085690, −8.240285469156036787224239706947, −7.973527302883012086253803309130, −7.948809127331393182790947574353, −7.18998231025423721057059653219, −6.96187275880324967937261143524, −6.65900584825536672477483208092, −5.93283270669073300636168200863, −5.80540908672813938371068804019, −5.54717372055707994807536643539, −4.76315528153700755702113643491, −4.72363048351793397227652307916, −4.08998623178261238037608116234, −3.66744408406304514006563530730, −3.37831701470308807991526259698, −2.89035302814761694590035412391, −2.23339281672873785482519299084, −1.76320036686752856954460013007, −0.909620922442004556223044632987, −0.71973304235221683152050880310, 0.71973304235221683152050880310, 0.909620922442004556223044632987, 1.76320036686752856954460013007, 2.23339281672873785482519299084, 2.89035302814761694590035412391, 3.37831701470308807991526259698, 3.66744408406304514006563530730, 4.08998623178261238037608116234, 4.72363048351793397227652307916, 4.76315528153700755702113643491, 5.54717372055707994807536643539, 5.80540908672813938371068804019, 5.93283270669073300636168200863, 6.65900584825536672477483208092, 6.96187275880324967937261143524, 7.18998231025423721057059653219, 7.948809127331393182790947574353, 7.973527302883012086253803309130, 8.240285469156036787224239706947, 8.911949685783390555619457085690

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