Properties

Label 4-2352e2-1.1-c1e2-0-18
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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 + 3·9-s + 10·19-s − 2·25-s − 4·27-s − 10·31-s − 22·37-s − 12·47-s + 24·53-s − 20·57-s + 24·59-s + 4·75-s + 5·81-s + 36·83-s + 20·93-s − 10·103-s − 14·109-s + 44·111-s − 12·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 24·141-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 2.29·19-s − 2/5·25-s − 0.769·27-s − 1.79·31-s − 3.61·37-s − 1.75·47-s + 3.29·53-s − 2.64·57-s + 3.12·59-s + 0.461·75-s + 5/9·81-s + 3.95·83-s + 2.07·93-s − 0.985·103-s − 1.34·109-s + 4.17·111-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.02·141-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.316937982\)
\(L(\frac12)\) \(\approx\) \(1.316937982\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 59 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 119 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
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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

−9.276721696535111346409947684709, −8.856140010917564065263200027539, −8.341952303918791598209048671146, −8.045962385903677579649840497420, −7.48424961501383884114538853338, −7.04840591138849773268916068226, −6.85976078204587706182350583504, −6.74612151642157778583310294098, −5.75565775758836832693955414795, −5.60324884093449673494405442307, −5.21499812717345520322333342865, −5.19485696281069393294831749671, −4.46243560852116920068933041513, −3.76191126042707419299838809967, −3.56493552501882482018495894551, −3.20838385293675385367088968887, −2.16570345940955167125357545982, −1.89013643503359349959142638018, −1.10397140458889933376716455222, −0.48343160271726237780202282251, 0.48343160271726237780202282251, 1.10397140458889933376716455222, 1.89013643503359349959142638018, 2.16570345940955167125357545982, 3.20838385293675385367088968887, 3.56493552501882482018495894551, 3.76191126042707419299838809967, 4.46243560852116920068933041513, 5.19485696281069393294831749671, 5.21499812717345520322333342865, 5.60324884093449673494405442307, 5.75565775758836832693955414795, 6.74612151642157778583310294098, 6.85976078204587706182350583504, 7.04840591138849773268916068226, 7.48424961501383884114538853338, 8.045962385903677579649840497420, 8.341952303918791598209048671146, 8.856140010917564065263200027539, 9.276721696535111346409947684709

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