Properties

Label 4-28e4-1.1-c3e2-0-11
Degree $4$
Conductor $614656$
Sign $1$
Analytic cond. $2139.75$
Root an. cond. $6.80128$
Motivic weight $3$
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  − 4·9-s + 28·11-s − 280·23-s + 142·25-s − 572·29-s − 76·37-s + 68·43-s − 148·53-s − 1.36e3·67-s − 1.17e3·71-s − 2.44e3·79-s − 713·81-s − 112·99-s + 3.36e3·107-s − 1.63e3·109-s − 1.08e3·113-s − 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.80e3·169-s + ⋯
L(s)  = 1  − 0.148·9-s + 0.767·11-s − 2.53·23-s + 1.13·25-s − 3.66·29-s − 0.337·37-s + 0.241·43-s − 0.383·53-s − 2.49·67-s − 1.96·71-s − 3.47·79-s − 0.978·81-s − 0.113·99-s + 3.04·107-s − 1.43·109-s − 0.899·113-s − 1.55·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.820·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(614656\)    =    \(2^{8} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2139.75\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{784} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 614656,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad2 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 142 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 14 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 1802 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9824 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 13716 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 140 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 286 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 50870 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 38 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 122000 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 66154 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 74 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 222260 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 453762 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 + 684 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 + 588 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 705072 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 1220 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 964772 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 1028000 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 - 375456 T^{2} + p^{6} T^{4} \)
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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

−9.684022984397692245454261874020, −9.128411349068611003610056321677, −8.968678444054366104727739920644, −8.527507421274420929489114186294, −7.88210339607289002893571701690, −7.43646476898576017409533630218, −7.30859896895116771811050638705, −6.58593248224015170219759449413, −6.07977814132360318667575791600, −5.67138315321316035460112516126, −5.48367601410815890892870391696, −4.46276853386934438296346152520, −4.25213160156057497604573999231, −3.71063663694475051529453530851, −3.17908902205713486983845163468, −2.47013692941292064025370315172, −1.65669549311643921673767257976, −1.48133480202169202166090406584, 0, 0, 1.48133480202169202166090406584, 1.65669549311643921673767257976, 2.47013692941292064025370315172, 3.17908902205713486983845163468, 3.71063663694475051529453530851, 4.25213160156057497604573999231, 4.46276853386934438296346152520, 5.48367601410815890892870391696, 5.67138315321316035460112516126, 6.07977814132360318667575791600, 6.58593248224015170219759449413, 7.30859896895116771811050638705, 7.43646476898576017409533630218, 7.88210339607289002893571701690, 8.527507421274420929489114186294, 8.968678444054366104727739920644, 9.128411349068611003610056321677, 9.684022984397692245454261874020

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