Properties

Label 4-495e2-1.1-c1e2-0-8
Degree $4$
Conductor $245025$
Sign $1$
Analytic cond. $15.6230$
Root an. cond. $1.98811$
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  − 4-s + 2·5-s + 4·7-s + 2·11-s + 4·13-s − 3·16-s + 4·19-s − 2·20-s + 3·25-s − 4·28-s − 8·31-s + 8·35-s + 4·37-s + 4·43-s − 2·44-s − 2·49-s − 4·52-s + 12·53-s + 4·55-s + 4·61-s + 7·64-s + 8·65-s + 16·67-s + 4·73-s − 4·76-s + 8·77-s − 20·79-s + ⋯
L(s)  = 1  − 1/2·4-s + 0.894·5-s + 1.51·7-s + 0.603·11-s + 1.10·13-s − 3/4·16-s + 0.917·19-s − 0.447·20-s + 3/5·25-s − 0.755·28-s − 1.43·31-s + 1.35·35-s + 0.657·37-s + 0.609·43-s − 0.301·44-s − 2/7·49-s − 0.554·52-s + 1.64·53-s + 0.539·55-s + 0.512·61-s + 7/8·64-s + 0.992·65-s + 1.95·67-s + 0.468·73-s − 0.458·76-s + 0.911·77-s − 2.25·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(245025\)    =    \(3^{4} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(15.6230\)
Root analytic conductor: \(1.98811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{495} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 245025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.479700196\)
\(L(\frac12)\) \(\approx\) \(2.479700196\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−11.30304771462910117059273356339, −10.89127882137597332044623208978, −10.20272727305579513766541437459, −9.817864852424828595250635592134, −9.235840824968697107592518496619, −9.084569257831171498526330273772, −8.371418722385280306901806747407, −8.312340729611254082072872823662, −7.57348372324036685001824067013, −7.00778467808788486620120121749, −6.64737652328587879582815975106, −5.90897232901399477075700043561, −5.35119451808805783258496348533, −5.29469398389400579696305945589, −4.29056875685508697222209046026, −4.17158148826676126371529980173, −3.31455491747336655126805878644, −2.42160994809114831712228434286, −1.69968450584551669242086668275, −1.10925676578488351934899627340, 1.10925676578488351934899627340, 1.69968450584551669242086668275, 2.42160994809114831712228434286, 3.31455491747336655126805878644, 4.17158148826676126371529980173, 4.29056875685508697222209046026, 5.29469398389400579696305945589, 5.35119451808805783258496348533, 5.90897232901399477075700043561, 6.64737652328587879582815975106, 7.00778467808788486620120121749, 7.57348372324036685001824067013, 8.312340729611254082072872823662, 8.371418722385280306901806747407, 9.084569257831171498526330273772, 9.235840824968697107592518496619, 9.817864852424828595250635592134, 10.20272727305579513766541437459, 10.89127882137597332044623208978, 11.30304771462910117059273356339

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