Properties

Label 4-6300e2-1.1-c1e2-0-13
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  − 8·11-s + 12·19-s + 4·31-s − 4·41-s − 49-s + 8·59-s − 4·61-s − 16·71-s + 16·79-s + 20·89-s + 36·101-s + 36·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  − 2.41·11-s + 2.75·19-s + 0.718·31-s − 0.624·41-s − 1/7·49-s + 1.04·59-s − 0.512·61-s − 1.89·71-s + 1.80·79-s + 2.11·89-s + 3.58·101-s + 3.44·109-s + 2.36·121-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-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

\(L(1)\) \(\approx\) \(2.817025392\)
\(L(\frac12)\) \(\approx\) \(2.817025392\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \)
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

−7.990559662677144569670114834507, −7.63385931608672609073757074326, −7.57592220008772355429940860215, −7.57582658508288949412195695933, −6.69122852851910054939075243518, −6.61066790558613938560169599183, −5.96966274657209857681384518013, −5.63450017220924607847725922146, −5.26219716224466096400241318443, −5.16110622370340951850376996907, −4.67698455715596792613815301249, −4.42633239557722465781627911247, −3.55169344196006477157150420368, −3.41507994243616603850191752731, −2.84100214533021432031491183088, −2.81390347650719663174981781874, −2.03265705781497066910104140023, −1.73526457130475088793701818690, −0.74731136274718146698992493677, −0.60054541029827081180135457380, 0.60054541029827081180135457380, 0.74731136274718146698992493677, 1.73526457130475088793701818690, 2.03265705781497066910104140023, 2.81390347650719663174981781874, 2.84100214533021432031491183088, 3.41507994243616603850191752731, 3.55169344196006477157150420368, 4.42633239557722465781627911247, 4.67698455715596792613815301249, 5.16110622370340951850376996907, 5.26219716224466096400241318443, 5.63450017220924607847725922146, 5.96966274657209857681384518013, 6.61066790558613938560169599183, 6.69122852851910054939075243518, 7.57582658508288949412195695933, 7.57592220008772355429940860215, 7.63385931608672609073757074326, 7.990559662677144569670114834507

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