Properties

Label 4-950e2-1.1-c1e2-0-5
Degree $4$
Conductor $902500$
Sign $1$
Analytic cond. $57.5441$
Root an. cond. $2.75423$
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 + 5·9-s + 4·11-s + 16-s + 2·19-s + 10·29-s − 16·31-s − 5·36-s − 16·41-s − 4·44-s + 5·49-s − 30·59-s + 4·61-s − 64-s + 4·71-s − 2·76-s + 20·79-s + 16·81-s + 20·99-s + 4·101-s + 30·109-s − 10·116-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 1.20·11-s + 1/4·16-s + 0.458·19-s + 1.85·29-s − 2.87·31-s − 5/6·36-s − 2.49·41-s − 0.603·44-s + 5/7·49-s − 3.90·59-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 0.229·76-s + 2.25·79-s + 16/9·81-s + 2.01·99-s + 0.398·101-s + 2.87·109-s − 0.928·116-s − 0.909·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(902500\)    =    \(2^{2} \cdot 5^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(57.5441\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{950} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 902500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.220223572\)
\(L(\frac12)\) \(\approx\) \(2.220223572\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
19$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 105 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
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

−10.20686177070777993702787758356, −9.896540214040990582267120060212, −9.290777812594767207445269516254, −8.993274606774424378511424752687, −8.921527482239301520337262919554, −7.919176603635043067459351098544, −7.908104386178803227334811603934, −7.18071432601313327592947201680, −6.92435317740451826562152346022, −6.47540811791126218630180644646, −6.08546600481128978129866741152, −5.22989644326966415173770998739, −5.06327090117280067815267076119, −4.33044112260669478401994054540, −4.16604537503819153879657329221, −3.35882232367253671756552109964, −3.24569016120247373212380306101, −1.79230948075124349973494254521, −1.73532994630650044859878851120, −0.75062600675756926202476192265, 0.75062600675756926202476192265, 1.73532994630650044859878851120, 1.79230948075124349973494254521, 3.24569016120247373212380306101, 3.35882232367253671756552109964, 4.16604537503819153879657329221, 4.33044112260669478401994054540, 5.06327090117280067815267076119, 5.22989644326966415173770998739, 6.08546600481128978129866741152, 6.47540811791126218630180644646, 6.92435317740451826562152346022, 7.18071432601313327592947201680, 7.908104386178803227334811603934, 7.919176603635043067459351098544, 8.921527482239301520337262919554, 8.993274606774424378511424752687, 9.290777812594767207445269516254, 9.896540214040990582267120060212, 10.20686177070777993702787758356

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