Properties

Label 4-3240e2-1.1-c1e2-0-18
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  − 5-s − 2·7-s + 2·11-s − 4·13-s + 4·17-s + 8·19-s + 8·23-s − 10·29-s − 4·31-s + 2·35-s + 8·43-s + 8·47-s + 7·49-s − 12·53-s − 2·55-s − 14·59-s + 14·61-s + 4·65-s + 4·67-s − 24·71-s + 12·73-s − 4·77-s + 12·79-s + 4·83-s − 4·85-s + 24·89-s + 8·91-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 0.603·11-s − 1.10·13-s + 0.970·17-s + 1.83·19-s + 1.66·23-s − 1.85·29-s − 0.718·31-s + 0.338·35-s + 1.21·43-s + 1.16·47-s + 49-s − 1.64·53-s − 0.269·55-s − 1.82·59-s + 1.79·61-s + 0.496·65-s + 0.488·67-s − 2.84·71-s + 1.40·73-s − 0.455·77-s + 1.35·79-s + 0.439·83-s − 0.433·85-s + 2.54·89-s + 0.838·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3240} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.461697488\)
\(L(\frac12)\) \(\approx\) \(2.461697488\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 8 T + 41 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
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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

−8.869462372282295202957172107483, −8.742890953810567990407208661612, −7.71447420287431932324308901774, −7.63004399668099205975332266263, −7.37451708309472897532855515521, −7.29406430226838872347904809550, −6.62723176863843212535958603291, −6.15890544092952890365779185306, −5.83200602809844329235036109013, −5.39802955626714196967458616118, −5.04366737560992184796880257472, −4.66963632843920062930073822790, −4.15047594691703027703477778111, −3.48974079891533671451568922716, −3.26061608063526723867505453473, −3.16561483306824179826255778109, −2.24450231914837688340994498212, −1.87727151863891096223848118861, −0.915317226796472180235243374418, −0.62823639313851453459713161561, 0.62823639313851453459713161561, 0.915317226796472180235243374418, 1.87727151863891096223848118861, 2.24450231914837688340994498212, 3.16561483306824179826255778109, 3.26061608063526723867505453473, 3.48974079891533671451568922716, 4.15047594691703027703477778111, 4.66963632843920062930073822790, 5.04366737560992184796880257472, 5.39802955626714196967458616118, 5.83200602809844329235036109013, 6.15890544092952890365779185306, 6.62723176863843212535958603291, 7.29406430226838872347904809550, 7.37451708309472897532855515521, 7.63004399668099205975332266263, 7.71447420287431932324308901774, 8.742890953810567990407208661612, 8.869462372282295202957172107483

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