Properties

Label 4-7e4-1.1-c3e2-0-2
Degree $4$
Conductor $2401$
Sign $1$
Analytic cond. $8.35842$
Root an. cond. $1.70032$
Motivic weight $3$
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  + 5·2-s + 8·4-s − 5·8-s + 27·9-s + 68·11-s − 25·16-s + 135·18-s + 340·22-s + 40·23-s + 125·25-s − 332·29-s − 40·32-s + 216·36-s − 450·37-s − 360·43-s + 544·44-s + 200·46-s + 625·50-s − 590·53-s − 1.66e3·58-s − 487·64-s + 740·67-s + 1.37e3·71-s − 135·72-s − 2.25e3·74-s + 1.38e3·79-s − 1.80e3·86-s + ⋯
L(s)  = 1  + 1.76·2-s + 4-s − 0.220·8-s + 9-s + 1.86·11-s − 0.390·16-s + 1.76·18-s + 3.29·22-s + 0.362·23-s + 25-s − 2.12·29-s − 0.220·32-s + 36-s − 1.99·37-s − 1.27·43-s + 1.86·44-s + 0.641·46-s + 1.76·50-s − 1.52·53-s − 3.75·58-s − 0.951·64-s + 1.34·67-s + 2.30·71-s − 0.220·72-s − 3.53·74-s + 1.97·79-s − 2.25·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $1$
Analytic conductor: \(8.35842\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{49} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2401,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.968010391\)
\(L(\frac12)\) \(\approx\) \(3.968010391\)
\(L(\frac{5}{2})\) 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 \( 1 \)
good2$C_2^2$ \( 1 - 5 T + 17 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \)
3$C_2$ \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \)
5$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 68 T + 3293 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 40 T - 10567 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 166 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 450 T + 151847 T^{2} + 450 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 180 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 590 T + 199223 T^{2} + 590 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 740 T + 246837 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 688 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1384 T + 1422417 T^{2} - 1384 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
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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

−15.58150961034859080593446467880, −14.56790425228937279716035732933, −14.26829920098738522370876216374, −13.71392684831262013852724372012, −13.14577009457773151276223702368, −12.57030356794099798553145416739, −12.35158545687795217665964664544, −11.52360495984211484578701326346, −10.99618450099550377412016358504, −10.09827857541056548882226540943, −9.306031528502591944840417068787, −8.935841680855088529441486642920, −7.79471782090051441495854058209, −6.80492720372833074655131100255, −6.51814749140058946600446650259, −5.32295841663858738679933007277, −4.84578728785744800676254557824, −3.79768215730231864999750578192, −3.61191549279543295011614079246, −1.61408681898428496179850998824, 1.61408681898428496179850998824, 3.61191549279543295011614079246, 3.79768215730231864999750578192, 4.84578728785744800676254557824, 5.32295841663858738679933007277, 6.51814749140058946600446650259, 6.80492720372833074655131100255, 7.79471782090051441495854058209, 8.935841680855088529441486642920, 9.306031528502591944840417068787, 10.09827857541056548882226540943, 10.99618450099550377412016358504, 11.52360495984211484578701326346, 12.35158545687795217665964664544, 12.57030356794099798553145416739, 13.14577009457773151276223702368, 13.71392684831262013852724372012, 14.26829920098738522370876216374, 14.56790425228937279716035732933, 15.58150961034859080593446467880

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