Properties

Label 4-90e4-1.1-c1e2-0-5
Degree $4$
Conductor $65610000$
Sign $1$
Analytic cond. $4183.35$
Root an. cond. $8.04231$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·11-s − 10·19-s + 18·29-s + 10·31-s − 18·41-s − 2·49-s − 18·59-s − 20·61-s + 6·71-s + 8·79-s + 18·89-s − 30·101-s − 22·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 1.80·11-s − 2.29·19-s + 3.34·29-s + 1.79·31-s − 2.81·41-s − 2/7·49-s − 2.34·59-s − 2.56·61-s + 0.712·71-s + 0.900·79-s + 1.90·89-s − 2.98·101-s − 2.10·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65610000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(65610000\)    =    \(2^{4} \cdot 3^{8} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(4183.35\)
Root analytic conductor: \(8.04231\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 65610000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.496638925\)
\(L(\frac12)\) \(\approx\) \(2.496638925\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.014314369223243594461048982039, −7.896991764601046083267678194026, −7.16367256370461133833970837946, −6.60597526108958956137347048781, −6.53239127785860156602342407887, −6.51472452494962458507393716037, −6.23906877904912093020757912163, −5.64383531484979410889158488991, −4.95059647334067404023635441222, −4.85091576748896371807942654703, −4.38374202551815685015196066995, −4.24118536048522755757380064591, −3.81109860777580603617551319637, −3.16774684067533091896441644117, −2.94407829053737253396460231590, −2.57730295356432369557971685450, −1.71680167446038212206007107819, −1.67560496926417688775667522351, −1.05487103233014165842616105893, −0.39933647876750581816733663898, 0.39933647876750581816733663898, 1.05487103233014165842616105893, 1.67560496926417688775667522351, 1.71680167446038212206007107819, 2.57730295356432369557971685450, 2.94407829053737253396460231590, 3.16774684067533091896441644117, 3.81109860777580603617551319637, 4.24118536048522755757380064591, 4.38374202551815685015196066995, 4.85091576748896371807942654703, 4.95059647334067404023635441222, 5.64383531484979410889158488991, 6.23906877904912093020757912163, 6.51472452494962458507393716037, 6.53239127785860156602342407887, 6.60597526108958956137347048781, 7.16367256370461133833970837946, 7.896991764601046083267678194026, 8.014314369223243594461048982039

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