Properties

Label 4-5445e2-1.1-c1e2-0-6
Degree $4$
Conductor $29648025$
Sign $1$
Analytic cond. $1890.38$
Root an. cond. $6.59382$
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  − 2·2-s + 4-s + 2·5-s + 4·7-s − 4·10-s − 8·14-s + 16-s − 8·17-s + 8·19-s + 2·20-s + 8·23-s + 3·25-s + 4·28-s − 4·29-s + 2·32-s + 16·34-s + 8·35-s + 12·37-s − 16·38-s + 4·41-s + 12·43-s − 16·46-s + 8·47-s + 6·49-s − 6·50-s + 4·53-s + 8·58-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 0.894·5-s + 1.51·7-s − 1.26·10-s − 2.13·14-s + 1/4·16-s − 1.94·17-s + 1.83·19-s + 0.447·20-s + 1.66·23-s + 3/5·25-s + 0.755·28-s − 0.742·29-s + 0.353·32-s + 2.74·34-s + 1.35·35-s + 1.97·37-s − 2.59·38-s + 0.624·41-s + 1.82·43-s − 2.35·46-s + 1.16·47-s + 6/7·49-s − 0.848·50-s + 0.549·53-s + 1.05·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29648025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29648025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29648025\)    =    \(3^{4} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1890.38\)
Root analytic conductor: \(6.59382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5445} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29648025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.313483270\)
\(L(\frac12)\) \(\approx\) \(2.313483270\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 12 T + 78 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.331016817388970988183371876315, −8.247678890309799175778647832503, −7.52826969760049550120180357149, −7.48925896783488924545672001575, −7.07342399351735640584013028888, −6.84122328744642054211108932895, −6.09585862356801641762218670768, −5.76370225568848786472270783883, −5.68263231865689236678490589789, −5.06886606576379787740390285001, −4.66881151650890024491299705208, −4.45812366000833001579529682874, −4.03369758628296482486033243557, −3.33987965375690973353145695154, −2.65799673594129466341023519489, −2.55876714030468699536870868796, −1.99360234043674250341966951984, −1.44067372505325557192109288454, −0.808556911048721832594774058780, −0.75955528901466713275009357434, 0.75955528901466713275009357434, 0.808556911048721832594774058780, 1.44067372505325557192109288454, 1.99360234043674250341966951984, 2.55876714030468699536870868796, 2.65799673594129466341023519489, 3.33987965375690973353145695154, 4.03369758628296482486033243557, 4.45812366000833001579529682874, 4.66881151650890024491299705208, 5.06886606576379787740390285001, 5.68263231865689236678490589789, 5.76370225568848786472270783883, 6.09585862356801641762218670768, 6.84122328744642054211108932895, 7.07342399351735640584013028888, 7.48925896783488924545672001575, 7.52826969760049550120180357149, 8.247678890309799175778647832503, 8.331016817388970988183371876315

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