Properties

Label 4-490e2-1.1-c1e2-0-19
Degree $4$
Conductor $240100$
Sign $-1$
Analytic cond. $15.3089$
Root an. cond. $1.97804$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 9-s + 16-s − 8·23-s − 25-s − 6·29-s − 36-s − 8·37-s − 12·43-s − 8·53-s + 64-s + 20·67-s − 4·71-s − 20·79-s − 8·81-s − 8·92-s − 100-s + 12·107-s − 2·109-s + 8·113-s − 6·116-s − 18·121-s + 127-s + 131-s + 137-s + 139-s − 144-s + ⋯
L(s)  = 1  + 1/2·4-s − 1/3·9-s + 1/4·16-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1/6·36-s − 1.31·37-s − 1.82·43-s − 1.09·53-s + 1/8·64-s + 2.44·67-s − 0.474·71-s − 2.25·79-s − 8/9·81-s − 0.834·92-s − 0.0999·100-s + 1.16·107-s − 0.191·109-s + 0.752·113-s − 0.557·116-s − 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(240100\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $-1$
Analytic conductor: \(15.3089\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 240100,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
5$C_2$ \( 1 + T^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 5 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
73$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 103 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 129 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
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

−8.678831705523340326545652473700, −8.224416239572792359446756199858, −7.84477582178301860280207781235, −7.33352489227896824982464612877, −6.74791701499940997518383557896, −6.44315517577070774610233060997, −5.71499191440762615317685767901, −5.49080639144099597772671774262, −4.79453071199600002822081908040, −4.07891098452839660483206604097, −3.55979711099246705792415994287, −2.98652964760512358792167314449, −2.09578724899994332343744223306, −1.62025429883584239177136663093, 0, 1.62025429883584239177136663093, 2.09578724899994332343744223306, 2.98652964760512358792167314449, 3.55979711099246705792415994287, 4.07891098452839660483206604097, 4.79453071199600002822081908040, 5.49080639144099597772671774262, 5.71499191440762615317685767901, 6.44315517577070774610233060997, 6.74791701499940997518383557896, 7.33352489227896824982464612877, 7.84477582178301860280207781235, 8.224416239572792359446756199858, 8.678831705523340326545652473700

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