Properties

Label 4-3360e2-1.1-c1e2-0-14
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
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  + 2·3-s + 2·5-s + 2·7-s + 3·9-s + 4·15-s + 4·21-s + 4·23-s + 3·25-s + 4·27-s + 8·29-s − 4·31-s + 4·35-s + 8·37-s + 4·41-s + 12·43-s + 6·45-s + 4·47-s + 3·49-s + 4·53-s − 8·59-s + 16·61-s + 6·63-s − 4·67-s + 8·69-s + 4·71-s + 8·73-s + 6·75-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s + 1.03·15-s + 0.872·21-s + 0.834·23-s + 3/5·25-s + 0.769·27-s + 1.48·29-s − 0.718·31-s + 0.676·35-s + 1.31·37-s + 0.624·41-s + 1.82·43-s + 0.894·45-s + 0.583·47-s + 3/7·49-s + 0.549·53-s − 1.04·59-s + 2.04·61-s + 0.755·63-s − 0.488·67-s + 0.963·69-s + 0.474·71-s + 0.936·73-s + 0.692·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.818369151\)
\(L(\frac12)\) \(\approx\) \(7.818369151\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_4$ \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \)
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

−8.667615124910568525897941137695, −8.664388061843238189325753064565, −7.942008096352279960572492571844, −7.894279060320489019889263271089, −7.28437247278640288380444857616, −7.15824262205076958805389537432, −6.54695511885551053288807143244, −6.26912723110204024189800012017, −5.70349654199735185466378954733, −5.47678973147421015334903150988, −4.74949927314861669090634041346, −4.74160312367346261047832282929, −4.04236370906893394626285276808, −3.81077979732484720244751559696, −3.08688408675868740199599919880, −2.68579138043797739454868308621, −2.33493158427018902618279765864, −1.98094352217896926746080943612, −1.09837937907201910720734104826, −0.952004215443621182430233406141, 0.952004215443621182430233406141, 1.09837937907201910720734104826, 1.98094352217896926746080943612, 2.33493158427018902618279765864, 2.68579138043797739454868308621, 3.08688408675868740199599919880, 3.81077979732484720244751559696, 4.04236370906893394626285276808, 4.74160312367346261047832282929, 4.74949927314861669090634041346, 5.47678973147421015334903150988, 5.70349654199735185466378954733, 6.26912723110204024189800012017, 6.54695511885551053288807143244, 7.15824262205076958805389537432, 7.28437247278640288380444857616, 7.894279060320489019889263271089, 7.942008096352279960572492571844, 8.664388061843238189325753064565, 8.667615124910568525897941137695

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