Properties

Label 4-30e4-1.1-c2e2-0-1
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $601.388$
Root an. cond. $4.95209$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 14·13-s − 14·19-s + 34·31-s − 32·37-s − 110·43-s − 95·49-s + 130·61-s − 98·67-s − 176·73-s − 80·79-s + 28·91-s + 82·97-s − 164·103-s − 98·109-s + 224·121-s + 127-s + 131-s + 28·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + ⋯
L(s)  = 1  − 2/7·7-s − 1.07·13-s − 0.736·19-s + 1.09·31-s − 0.864·37-s − 2.55·43-s − 1.93·49-s + 2.13·61-s − 1.46·67-s − 2.41·73-s − 1.01·79-s + 4/13·91-s + 0.845·97-s − 1.59·103-s − 0.899·109-s + 1.85·121-s + 0.00787·127-s + 0.00763·131-s + 4/19·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(601.388\)
Root analytic conductor: \(4.95209\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 810000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6337478220\)
\(L(\frac12)\) \(\approx\) \(0.6337478220\)
\(L(2)\) 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 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 + T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 224 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 560 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 + 7 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 176 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 800 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 17 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 770 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 55 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 2240 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 + 1582 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 3920 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 65 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 49 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7490 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 88 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 40 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 10864 T^{2} + p^{4} T^{4} \)
89$C_2$ \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \)
97$C_2$ \( ( 1 - 41 T + p^{2} T^{2} )^{2} \)
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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

−10.21905658310920424624800611829, −9.873532653723532485609668440358, −9.351530177106777895322213775939, −8.698942946515618136103189578833, −8.583328510590130601161354482856, −8.026068277993017347704229638670, −7.64503409001345623449333398466, −6.95618777685981261028276547107, −6.83270660119798512929282084297, −6.30727035105721454465504149852, −5.84316269493855837198368425176, −5.08554236692868200805494491873, −4.97677967019516916225446648597, −4.32692976011623481158418626891, −3.83394919632874867023228838620, −3.05219076596568358860490604859, −2.82307665051756833932936444782, −1.95701962426430871495074973759, −1.44155157006035149366648738820, −0.25413368709989913992283162752, 0.25413368709989913992283162752, 1.44155157006035149366648738820, 1.95701962426430871495074973759, 2.82307665051756833932936444782, 3.05219076596568358860490604859, 3.83394919632874867023228838620, 4.32692976011623481158418626891, 4.97677967019516916225446648597, 5.08554236692868200805494491873, 5.84316269493855837198368425176, 6.30727035105721454465504149852, 6.83270660119798512929282084297, 6.95618777685981261028276547107, 7.64503409001345623449333398466, 8.026068277993017347704229638670, 8.583328510590130601161354482856, 8.698942946515618136103189578833, 9.351530177106777895322213775939, 9.873532653723532485609668440358, 10.21905658310920424624800611829

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