Properties

Label 4-847e2-1.1-c1e2-0-11
Degree $4$
Conductor $717409$
Sign $1$
Analytic cond. $45.7426$
Root an. cond. $2.60064$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 3-s + 4·4-s + 2·5-s + 3·6-s + 2·7-s − 3·8-s − 4·9-s − 6·10-s − 4·12-s + 2·13-s − 6·14-s − 2·15-s + 3·16-s − 5·17-s + 12·18-s − 8·19-s + 8·20-s − 2·21-s − 23-s + 3·24-s − 7·25-s − 6·26-s + 6·27-s + 8·28-s − 7·29-s + 6·30-s + ⋯
L(s)  = 1  − 2.12·2-s − 0.577·3-s + 2·4-s + 0.894·5-s + 1.22·6-s + 0.755·7-s − 1.06·8-s − 4/3·9-s − 1.89·10-s − 1.15·12-s + 0.554·13-s − 1.60·14-s − 0.516·15-s + 3/4·16-s − 1.21·17-s + 2.82·18-s − 1.83·19-s + 1.78·20-s − 0.436·21-s − 0.208·23-s + 0.612·24-s − 7/5·25-s − 1.17·26-s + 1.15·27-s + 1.51·28-s − 1.29·29-s + 1.09·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(717409\)    =    \(7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(45.7426\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{847} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 717409,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 - 43 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T - 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 11 T + 123 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 13 T + 163 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 24 T + 285 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 15 T + 213 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
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

−9.865078960704782089208818551945, −9.482838818025454908029145363388, −8.984160913290049994533083852919, −8.876017411837362029824939682435, −8.366164039787206267798900311225, −8.191099614969721863271154062532, −7.61495633123750741670707522513, −7.22059825385241821925502660797, −6.39173109338585073287907499930, −6.05112249494461483758841946808, −5.97163257254285904479895135582, −5.30398997807561911981799181329, −4.67318066006327461208361749261, −4.11489933847878249489477622553, −3.33236426741763899825414396496, −2.48188325004143288463047994564, −1.85083387354783508676328934534, −1.55579214088232074399455203926, 0, 0, 1.55579214088232074399455203926, 1.85083387354783508676328934534, 2.48188325004143288463047994564, 3.33236426741763899825414396496, 4.11489933847878249489477622553, 4.67318066006327461208361749261, 5.30398997807561911981799181329, 5.97163257254285904479895135582, 6.05112249494461483758841946808, 6.39173109338585073287907499930, 7.22059825385241821925502660797, 7.61495633123750741670707522513, 8.191099614969721863271154062532, 8.366164039787206267798900311225, 8.876017411837362029824939682435, 8.984160913290049994533083852919, 9.482838818025454908029145363388, 9.865078960704782089208818551945

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