Properties

Label 4-1840e2-1.1-c3e2-0-0
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $11786.0$
Root an. cond. $10.4193$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 10·5-s + 17·7-s − 29·9-s − 18·11-s − 9·13-s + 30·15-s − 79·17-s − 34·19-s + 51·21-s + 46·23-s + 75·25-s − 120·27-s − 112·29-s + 92·31-s − 54·33-s + 170·35-s − 491·37-s − 27·39-s − 332·41-s + 354·43-s − 290·45-s + 599·47-s − 451·49-s − 237·51-s − 305·53-s − 180·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s + 0.917·7-s − 1.07·9-s − 0.493·11-s − 0.192·13-s + 0.516·15-s − 1.12·17-s − 0.410·19-s + 0.529·21-s + 0.417·23-s + 3/5·25-s − 0.855·27-s − 0.717·29-s + 0.533·31-s − 0.284·33-s + 0.821·35-s − 2.18·37-s − 0.110·39-s − 1.26·41-s + 1.25·43-s − 0.960·45-s + 1.85·47-s − 1.31·49-s − 0.650·51-s − 0.790·53-s − 0.441·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(11786.0\)
Root analytic conductor: \(10.4193\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3385600,\ (\ :3/2, 3/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - p T )^{2} \)
23$C_1$ \( ( 1 - p T )^{2} \)
good3$D_{4}$ \( 1 - p T + 38 T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 - 17 T + 740 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 18 T + 918 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 2936 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 79 T + 9178 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 34 T + 12182 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 112 T + 51841 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 92 T + 49361 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 491 T + 160682 T^{2} + 491 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 332 T + 65461 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 354 T + 181510 T^{2} - 354 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 599 T + 296890 T^{2} - 599 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 305 T + 320116 T^{2} + 305 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 357 T + 241414 T^{2} + 357 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 172 T + 458730 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 531 T + 641338 T^{2} - 531 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1254 T + 911559 T^{2} + 1254 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 343 T + 797792 T^{2} + 343 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 88 T + 930782 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 1273 T + 1298882 T^{2} + 1273 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1106 T + 1709834 T^{2} - 1106 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 2240 T + 2588894 T^{2} + 2240 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.658651984570933042072616875819, −8.532191546852113821629679138409, −7.976400219246335138037263665791, −7.60362378178508773517405898042, −7.07731556967921706899721246511, −6.84479690299551896496134151858, −6.08256747155476548723584700313, −6.00539271498279358544743480076, −5.34088831605711140028094693911, −5.16808344007236317121424001798, −4.54244723095298727619703873010, −4.33503444431509429041727807357, −3.39176712738665512180738419409, −3.21283793617400360688332770152, −2.39516538415409191031470514782, −2.35314512343187534487665146422, −1.67709716860612405566624717637, −1.22891382027858587386050653617, 0, 0, 1.22891382027858587386050653617, 1.67709716860612405566624717637, 2.35314512343187534487665146422, 2.39516538415409191031470514782, 3.21283793617400360688332770152, 3.39176712738665512180738419409, 4.33503444431509429041727807357, 4.54244723095298727619703873010, 5.16808344007236317121424001798, 5.34088831605711140028094693911, 6.00539271498279358544743480076, 6.08256747155476548723584700313, 6.84479690299551896496134151858, 7.07731556967921706899721246511, 7.60362378178508773517405898042, 7.976400219246335138037263665791, 8.532191546852113821629679138409, 8.658651984570933042072616875819

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