Properties

Label 4-600e2-1.1-c2e2-0-1
Degree $4$
Conductor $360000$
Sign $1$
Analytic cond. $267.283$
Root an. cond. $4.04336$
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  − 2·3-s + 12·7-s − 5·9-s − 20·13-s + 4·19-s − 24·21-s + 28·27-s − 44·31-s + 12·37-s + 40·39-s − 164·43-s + 10·49-s − 8·57-s − 172·61-s − 60·63-s − 4·67-s − 164·73-s + 20·79-s − 11·81-s − 240·91-s + 88·93-s + 188·97-s + 268·103-s + 20·109-s − 24·111-s + 100·117-s + 210·121-s + ⋯
L(s)  = 1  − 2/3·3-s + 12/7·7-s − 5/9·9-s − 1.53·13-s + 4/19·19-s − 8/7·21-s + 1.03·27-s − 1.41·31-s + 0.324·37-s + 1.02·39-s − 3.81·43-s + 0.204·49-s − 0.140·57-s − 2.81·61-s − 0.952·63-s − 0.0597·67-s − 2.24·73-s + 0.253·79-s − 0.135·81-s − 2.63·91-s + 0.946·93-s + 1.93·97-s + 2.60·103-s + 0.183·109-s − 0.216·111-s + 0.854·117-s + 1.73·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(267.283\)
Root analytic conductor: \(4.04336\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{600} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 360000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9642522982\)
\(L(\frac12)\) \(\approx\) \(0.9642522982\)
\(L(2)\) 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$C_2$ \( 1 + 2 T + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 210 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 10 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 66 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 930 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1394 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 2210 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 190 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 1746 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 1554 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 86 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 2 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 5406 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 + 82 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 8370 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 14690 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 - 94 T + p^{2} T^{2} )^{2} \)
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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

−10.70244785868761495917228250825, −10.17044126896561567826761969354, −10.12829414983079771637782043312, −9.194970347199334460215534819667, −9.053177211766155184430251379533, −8.333118716657602660312684393582, −8.105317707445382906380271735112, −7.43469559228312728443862544050, −7.36125129788992719779666357132, −6.55865421523715370040808558026, −6.14612600969401866276264379837, −5.43593877596170304469492259870, −5.14112255413689336688448336473, −4.70791545687891635746465022033, −4.45643443198741955659301751784, −3.32831464461184648419747785828, −2.96985268159926244867399890483, −1.81352770494242502634152029841, −1.73702581797980401787472406948, −0.37341211478045611868561431226, 0.37341211478045611868561431226, 1.73702581797980401787472406948, 1.81352770494242502634152029841, 2.96985268159926244867399890483, 3.32831464461184648419747785828, 4.45643443198741955659301751784, 4.70791545687891635746465022033, 5.14112255413689336688448336473, 5.43593877596170304469492259870, 6.14612600969401866276264379837, 6.55865421523715370040808558026, 7.36125129788992719779666357132, 7.43469559228312728443862544050, 8.105317707445382906380271735112, 8.333118716657602660312684393582, 9.053177211766155184430251379533, 9.194970347199334460215534819667, 10.12829414983079771637782043312, 10.17044126896561567826761969354, 10.70244785868761495917228250825

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