Properties

Label 4-6e8-1.1-c1e2-0-25
Degree $4$
Conductor $1679616$
Sign $1$
Analytic cond. $107.093$
Root an. cond. $3.21692$
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  + 5·7-s + 7·13-s + 2·19-s + 5·25-s − 4·31-s − 2·37-s + 8·43-s + 7·49-s + 13·61-s + 11·67-s + 34·73-s − 13·79-s + 35·91-s − 5·97-s − 7·103-s + 4·109-s + 11·121-s + 127-s + 131-s + 10·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.88·7-s + 1.94·13-s + 0.458·19-s + 25-s − 0.718·31-s − 0.328·37-s + 1.21·43-s + 49-s + 1.66·61-s + 1.34·67-s + 3.97·73-s − 1.46·79-s + 3.66·91-s − 0.507·97-s − 0.689·103-s + 0.383·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.867·133-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1679616\)    =    \(2^{8} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(107.093\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1296} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1679616,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.715723863\)
\(L(\frac12)\) \(\approx\) \(3.715723863\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} ) \)
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
37$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 17 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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

−9.611907970609641445728335825035, −9.605601286847581918190700150285, −8.764674666036761851865629304804, −8.666624470547837945098700561827, −8.280984797932974648309432722571, −7.947006811232314191575044139511, −7.50634895166366102485810172571, −7.04087245293201946205076287175, −6.51812222681218102349888219477, −6.18859073891285224981125564336, −5.50738124848484836225764434201, −5.19990589401770204189990802203, −4.93979110726106068881107107595, −4.21506353738067378167055571986, −3.80216600046040162357829510346, −3.46098609719550964888871724813, −2.57203772038842118031010330300, −2.04461999123608432021111609236, −1.30849831960923004611033906852, −0.967404041171581313564045931711, 0.967404041171581313564045931711, 1.30849831960923004611033906852, 2.04461999123608432021111609236, 2.57203772038842118031010330300, 3.46098609719550964888871724813, 3.80216600046040162357829510346, 4.21506353738067378167055571986, 4.93979110726106068881107107595, 5.19990589401770204189990802203, 5.50738124848484836225764434201, 6.18859073891285224981125564336, 6.51812222681218102349888219477, 7.04087245293201946205076287175, 7.50634895166366102485810172571, 7.947006811232314191575044139511, 8.280984797932974648309432722571, 8.666624470547837945098700561827, 8.764674666036761851865629304804, 9.605601286847581918190700150285, 9.611907970609641445728335825035

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