Properties

Label 4-6300e2-1.1-c1e2-0-5
Degree $4$
Conductor $39690000$
Sign $1$
Analytic cond. $2530.66$
Root an. cond. $7.09265$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·11-s + 8·19-s − 18·29-s − 4·31-s − 16·41-s − 49-s + 8·59-s + 8·61-s − 10·71-s + 30·79-s + 16·89-s − 12·101-s − 2·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 0.603·11-s + 1.83·19-s − 3.34·29-s − 0.718·31-s − 2.49·41-s − 1/7·49-s + 1.04·59-s + 1.02·61-s − 1.18·71-s + 3.37·79-s + 1.69·89-s − 1.19·101-s − 0.191·109-s − 1.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(39690000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(2530.66\)
Root analytic conductor: \(7.09265\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{6300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 39690000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8709250943\)
\(L(\frac12)\) \(\approx\) \(0.8709250943\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 53 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.065055704802852972657316644245, −7.78728129016738059251707013052, −7.64989451447186496444738119129, −7.02707082520411455053647391266, −7.00536013754967448993416803669, −6.57197336496874295420721794849, −5.95101393666717420279106847819, −5.61127602338234376591534919499, −5.42020368058318152062149337821, −4.96971586387074748007462735499, −4.91291604098597955527608232797, −4.06711893926834792073637116702, −3.63874775748121788207561582970, −3.51550291424625519016881134194, −3.17643738442983187785067292897, −2.36436942926770353028249561289, −2.17442420946082881633318094045, −1.58596022037096847735973572821, −1.11141398263344606846553954575, −0.24362805320620168716660852740, 0.24362805320620168716660852740, 1.11141398263344606846553954575, 1.58596022037096847735973572821, 2.17442420946082881633318094045, 2.36436942926770353028249561289, 3.17643738442983187785067292897, 3.51550291424625519016881134194, 3.63874775748121788207561582970, 4.06711893926834792073637116702, 4.91291604098597955527608232797, 4.96971586387074748007462735499, 5.42020368058318152062149337821, 5.61127602338234376591534919499, 5.95101393666717420279106847819, 6.57197336496874295420721794849, 7.00536013754967448993416803669, 7.02707082520411455053647391266, 7.64989451447186496444738119129, 7.78728129016738059251707013052, 8.065055704802852972657316644245

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