Properties

Label 4-9075e2-1.1-c1e2-0-11
Degree $4$
Conductor $82355625$
Sign $1$
Analytic cond. $5251.06$
Root an. cond. $8.51259$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4-s + 3·9-s − 2·12-s − 3·16-s + 12·23-s + 4·27-s + 8·31-s − 3·36-s + 22·37-s − 6·48-s − 2·49-s + 18·53-s − 12·59-s + 7·64-s + 4·67-s + 24·69-s − 12·71-s + 5·81-s + 18·89-s − 12·92-s + 16·93-s + 14·97-s − 28·103-s − 4·108-s + 44·111-s + 42·113-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s − 3/4·16-s + 2.50·23-s + 0.769·27-s + 1.43·31-s − 1/2·36-s + 3.61·37-s − 0.866·48-s − 2/7·49-s + 2.47·53-s − 1.56·59-s + 7/8·64-s + 0.488·67-s + 2.88·69-s − 1.42·71-s + 5/9·81-s + 1.90·89-s − 1.25·92-s + 1.65·93-s + 1.42·97-s − 2.75·103-s − 0.384·108-s + 4.17·111-s + 3.95·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(82355625\)    =    \(3^{2} \cdot 5^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(5251.06\)
Root analytic conductor: \(8.51259\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 82355625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.206157536\)
\(L(\frac12)\) \(\approx\) \(6.206157536\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
11 \( 1 \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 31 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 + 55 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−7.84605671860960842263485631020, −7.56626177808390425503802566062, −7.43292872227637892959162504220, −6.77263142478702577736009670529, −6.74544508007790591047393054058, −6.31257080390574645679109099904, −5.69709533654698555277739253123, −5.65654718699107272828865157544, −4.82935284143458366473936122396, −4.71837506330001720979415272207, −4.34908154766420695757309958265, −4.25481102489424678687943795489, −3.45900852155314022953775734056, −3.26783080166911657500046920693, −2.83839711710534354804617616625, −2.36809759390906627924853916185, −2.30672533754142861420352165627, −1.43276252738858661342632756536, −0.898254053855667962944737968862, −0.67152987784046778169232541463, 0.67152987784046778169232541463, 0.898254053855667962944737968862, 1.43276252738858661342632756536, 2.30672533754142861420352165627, 2.36809759390906627924853916185, 2.83839711710534354804617616625, 3.26783080166911657500046920693, 3.45900852155314022953775734056, 4.25481102489424678687943795489, 4.34908154766420695757309958265, 4.71837506330001720979415272207, 4.82935284143458366473936122396, 5.65654718699107272828865157544, 5.69709533654698555277739253123, 6.31257080390574645679109099904, 6.74544508007790591047393054058, 6.77263142478702577736009670529, 7.43292872227637892959162504220, 7.56626177808390425503802566062, 7.84605671860960842263485631020

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