Properties

Label 4-77e4-1.1-c1e2-0-7
Degree $4$
Conductor $35153041$
Sign $1$
Analytic cond. $2241.38$
Root an. cond. $6.88064$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 6-s + 8-s − 2·9-s − 6·13-s − 16-s − 9·17-s − 2·18-s + 6·19-s − 9·23-s + 24-s + 3·25-s − 6·26-s − 2·27-s + 13·29-s − 2·31-s − 6·32-s − 9·34-s + 4·37-s + 6·38-s − 6·39-s − 14·41-s + 7·43-s − 9·46-s − 7·47-s − 48-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.66·13-s − 1/4·16-s − 2.18·17-s − 0.471·18-s + 1.37·19-s − 1.87·23-s + 0.204·24-s + 3/5·25-s − 1.17·26-s − 0.384·27-s + 2.41·29-s − 0.359·31-s − 1.06·32-s − 1.54·34-s + 0.657·37-s + 0.973·38-s − 0.960·39-s − 2.18·41-s + 1.06·43-s − 1.32·46-s − 1.02·47-s − 0.144·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35153041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(35153041\)    =    \(7^{4} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2241.38\)
Root analytic conductor: \(6.88064\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5929} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 35153041,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$
bad7 \( 1 \)
11 \( 1 \)
good2$D_{4}$ \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 9 T + 3 p T^{2} + 9 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 + 9 T + 63 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 13 T + 97 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
43$D_{4}$ \( 1 - 7 T + 95 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 7 T + 77 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 17 T + 187 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 125 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + T + 53 T^{2} + p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 5 T + 145 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 3 T - 83 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 181 T^{2} + p^{2} T^{4} \)
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

−7.953355790671934742512563522500, −7.53740101524067991837167932917, −7.19186081668000685798879403949, −6.88401448939174499490817719172, −6.36546229577635044538328178787, −6.26928469160653902768183068485, −5.78891535535513383998126992435, −5.10432543421592402421766343065, −4.91481345673752603231065213657, −4.69766613703040098006786847398, −4.39196544169802577884306100993, −3.95704701578755866902869754945, −3.34450237365559872646035386774, −2.90458580459691654215486753120, −2.79919684210147768572098700246, −2.02706730455830808668465683113, −2.01454137764843544928418503701, −1.17395523329034865877634280036, 0, 0, 1.17395523329034865877634280036, 2.01454137764843544928418503701, 2.02706730455830808668465683113, 2.79919684210147768572098700246, 2.90458580459691654215486753120, 3.34450237365559872646035386774, 3.95704701578755866902869754945, 4.39196544169802577884306100993, 4.69766613703040098006786847398, 4.91481345673752603231065213657, 5.10432543421592402421766343065, 5.78891535535513383998126992435, 6.26928469160653902768183068485, 6.36546229577635044538328178787, 6.88401448939174499490817719172, 7.19186081668000685798879403949, 7.53740101524067991837167932917, 7.953355790671934742512563522500

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