Properties

Label 4-42e4-1.1-c3e2-0-25
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $10832.5$
Root an. cond. $10.2019$
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  − 3·5-s + 51·11-s − 61·13-s + 24·17-s − 169·19-s + 192·23-s − 115·25-s + 39·29-s + 92·31-s − 173·37-s + 174·41-s − 497·43-s + 180·47-s − 285·53-s − 153·55-s − 1.26e3·59-s − 328·61-s + 183·65-s − 875·67-s + 1.40e3·71-s + 1.36e3·73-s − 182·79-s − 399·83-s − 72·85-s + 822·89-s + 507·95-s − 841·97-s + ⋯
L(s)  = 1  − 0.268·5-s + 1.39·11-s − 1.30·13-s + 0.342·17-s − 2.04·19-s + 1.74·23-s − 0.919·25-s + 0.249·29-s + 0.533·31-s − 0.768·37-s + 0.662·41-s − 1.76·43-s + 0.558·47-s − 0.738·53-s − 0.375·55-s − 2.80·59-s − 0.688·61-s + 0.349·65-s − 1.59·67-s + 2.34·71-s + 2.18·73-s − 0.259·79-s − 0.527·83-s − 0.0918·85-s + 0.979·89-s + 0.547·95-s − 0.880·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(10832.5\)
Root analytic conductor: \(10.2019\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3111696,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5$D_{4}$ \( 1 + 3 T + 124 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 51 T + 3184 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 61 T + 2118 T^{2} + 61 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 24 T + 1762 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 169 T + 20730 T^{2} + 169 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2$ \( ( 1 - 96 T + p^{3} T^{2} )^{2} \)
29$D_{4}$ \( 1 - 39 T + 12094 T^{2} - 39 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 92 T + 48873 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 173 T + 62490 T^{2} + 173 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 174 T - 2846 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 497 T + 174468 T^{2} + 497 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 180 T + 182914 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 285 T + 224566 T^{2} + 285 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 1269 T + 13766 p T^{2} + 1269 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 328 T + 314646 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 875 T + 782544 T^{2} + 875 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1404 T + 1077298 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 1361 T + 1184556 T^{2} - 1361 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 182 T + 992307 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 399 T + 946240 T^{2} + 399 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 822 T + 1516786 T^{2} - 822 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 841 T + 1945608 T^{2} + 841 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.857059730149260481585356176890, −8.370677412707212591804217431868, −7.79494081819462023944810151523, −7.74938029389396300171112287861, −7.03343952562559409027075174097, −6.75871422246413622068724261968, −6.31227802091602216399225308992, −6.20943604888110246348196897130, −5.35934097507056167662978270457, −5.01678580370133048506206730101, −4.43740789026320731040443953417, −4.40759872487563625004749658267, −3.48422414253991090124521783926, −3.45659873066231321851475586909, −2.57154946108970791892650913271, −2.24499156618879278968416858530, −1.49098349108638408990294396944, −1.12304985242660808172740169271, 0, 0, 1.12304985242660808172740169271, 1.49098349108638408990294396944, 2.24499156618879278968416858530, 2.57154946108970791892650913271, 3.45659873066231321851475586909, 3.48422414253991090124521783926, 4.40759872487563625004749658267, 4.43740789026320731040443953417, 5.01678580370133048506206730101, 5.35934097507056167662978270457, 6.20943604888110246348196897130, 6.31227802091602216399225308992, 6.75871422246413622068724261968, 7.03343952562559409027075174097, 7.74938029389396300171112287861, 7.79494081819462023944810151523, 8.370677412707212591804217431868, 8.857059730149260481585356176890

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