Properties

Label 4-810e2-1.1-c3e2-0-30
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $2284.03$
Root an. cond. $6.91314$
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  − 4·2-s + 12·4-s − 10·5-s − 26·7-s − 32·8-s + 40·10-s + 36·11-s − 20·13-s + 104·14-s + 80·16-s + 90·17-s − 38·19-s − 120·20-s − 144·22-s + 120·23-s + 75·25-s + 80·26-s − 312·28-s + 324·29-s − 38·31-s − 192·32-s − 360·34-s + 260·35-s − 146·37-s + 152·38-s + 320·40-s + 252·41-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s − 1.40·7-s − 1.41·8-s + 1.26·10-s + 0.986·11-s − 0.426·13-s + 1.98·14-s + 5/4·16-s + 1.28·17-s − 0.458·19-s − 1.34·20-s − 1.39·22-s + 1.08·23-s + 3/5·25-s + 0.603·26-s − 2.10·28-s + 2.07·29-s − 0.220·31-s − 1.06·32-s − 1.81·34-s + 1.25·35-s − 0.648·37-s + 0.648·38-s + 1.26·40-s + 0.959·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2284.03\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 656100,\ (\ :3/2, 3/2),\ 1)\)

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$C_1$ \( ( 1 + p T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + p T )^{2} \)
good7$D_{4}$ \( 1 + 26 T + 852 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 36 T + 2623 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 20 T + 4482 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 90 T + 3424 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 2 p T + 13311 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 120 T + 26482 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 324 T + 72139 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 38 T + 58491 T^{2} + 38 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 146 T + 18912 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 252 T + 150451 T^{2} - 252 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 422 T + 182868 T^{2} + 422 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 366 T + 151348 T^{2} + 366 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 162 T + 303448 T^{2} + 162 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 72 T + 25411 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 368 T + 7650 p T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 + 296 T + 342522 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 240 T + 525859 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 710 T + 690192 T^{2} + 710 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 520 T + 921378 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 66 T + 186988 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 1248 T + 1797127 T^{2} - 1248 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1706 T + 2441208 T^{2} + 1706 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.654784027680891300284413634815, −9.237035354013312545208073483340, −8.833882971508811751012441259918, −8.474760376303520430259799653569, −7.86499473860894784317496061291, −7.75641872864496706117784577362, −6.97383211709163753213154007105, −6.79503371023917347511083923850, −6.33979153288240476984867438372, −6.10898350194349909846097721424, −5.02182504579917805493535106471, −4.90338378375115438072338388415, −3.73679292848377817719034226437, −3.66461886476298817424213252051, −2.78782056507827438949603926411, −2.75494438929662580148723244450, −1.26938845961446593578094342042, −1.26392919298259022391650211337, 0, 0, 1.26392919298259022391650211337, 1.26938845961446593578094342042, 2.75494438929662580148723244450, 2.78782056507827438949603926411, 3.66461886476298817424213252051, 3.73679292848377817719034226437, 4.90338378375115438072338388415, 5.02182504579917805493535106471, 6.10898350194349909846097721424, 6.33979153288240476984867438372, 6.79503371023917347511083923850, 6.97383211709163753213154007105, 7.75641872864496706117784577362, 7.86499473860894784317496061291, 8.474760376303520430259799653569, 8.833882971508811751012441259918, 9.237035354013312545208073483340, 9.654784027680891300284413634815

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