Properties

Label 4-117e2-1.1-c7e2-0-0
Degree $4$
Conductor $13689$
Sign $1$
Analytic cond. $1335.83$
Root an. cond. $6.04558$
Motivic weight $7$
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  + 128·4-s − 1.76e3·7-s + 1.26e4·13-s + 5.74e4·19-s − 1.56e5·25-s − 2.25e5·28-s − 6.62e5·31-s − 2.79e5·37-s + 6.25e5·43-s + 8.23e5·49-s + 1.61e6·52-s − 1.99e6·61-s − 2.09e6·64-s − 4.44e6·67-s + 1.00e7·73-s + 7.35e6·76-s − 9.03e6·79-s − 2.22e7·91-s + 1.75e7·97-s − 2.00e7·100-s − 4.38e7·103-s + 5.34e7·109-s + 1.94e7·121-s − 8.48e7·124-s + 127-s + 131-s − 1.01e8·133-s + ⋯
L(s)  = 1  + 4-s − 1.94·7-s + 1.59·13-s + 1.92·19-s − 2·25-s − 1.94·28-s − 3.99·31-s − 0.907·37-s + 1.20·43-s + 49-s + 1.59·52-s − 1.12·61-s − 64-s − 1.80·67-s + 3.03·73-s + 1.92·76-s − 2.06·79-s − 3.09·91-s + 1.94·97-s − 2·100-s − 3.95·103-s + 3.95·109-s + 121-s − 3.99·124-s − 3.73·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13689 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(13689\)    =    \(3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1335.83\)
Root analytic conductor: \(6.04558\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 13689,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.205701714\)
\(L(\frac12)\) \(\approx\) \(1.205701714\)
\(L(\frac{9}{2})\) 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)$
bad3 \( 1 \)
13$C_2$ \( 1 - 12605 T + p^{7} T^{2} \)
good2$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
5$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
7$C_2$ \( ( 1 + 508 T + p^{7} T^{2} )( 1 + 1255 T + p^{7} T^{2} ) \)
11$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
17$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
19$C_2$ \( ( 1 - 43091 T + p^{7} T^{2} )( 1 - 14357 T + p^{7} T^{2} ) \)
23$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
29$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
31$C_2$ \( ( 1 + 331387 T + p^{7} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 335663 T + p^{7} T^{2} )( 1 + 615373 T + p^{7} T^{2} ) \)
41$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
43$C_2$ \( ( 1 - 1035224 T + p^{7} T^{2} )( 1 + 409495 T + p^{7} T^{2} ) \)
47$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
59$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
61$C_2$ \( ( 1 - 1537199 T + p^{7} T^{2} )( 1 + 3535546 T + p^{7} T^{2} ) \)
67$C_2$ \( ( 1 + 385072 T + p^{7} T^{2} )( 1 + 4058455 T + p^{7} T^{2} ) \)
71$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
73$C_2$ \( ( 1 - 5038001 T + p^{7} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 4517617 T + p^{7} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{7} T^{2} )^{2} \)
89$C_2^2$ \( 1 - p^{7} T^{2} + p^{14} T^{4} \)
97$C_2$ \( ( 1 - 12245198 T + p^{7} T^{2} )( 1 - 5276357 T + p^{7} T^{2} ) \)
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

−12.61593972616659772037447655996, −11.73447576609460095821174292080, −11.47720137033973757827720181597, −10.81827676207360587660495376622, −10.50323342657472674293163692824, −9.512109378067277170689083727635, −9.458264031270819351117623266355, −8.925990233820996665478086720144, −7.927576317124303856636911842824, −7.17809003605299103764493011803, −7.16398485061566486905513472860, −6.12551683197131201048524442305, −5.95312127751398422826120290221, −5.37978071273508262860224714024, −3.97049350825627285084866584456, −3.42219124083318195414951729401, −3.18649607313584200768438525370, −2.05015100499069875518352988895, −1.45960808699802255086680717662, −0.30890239167143525101681596319, 0.30890239167143525101681596319, 1.45960808699802255086680717662, 2.05015100499069875518352988895, 3.18649607313584200768438525370, 3.42219124083318195414951729401, 3.97049350825627285084866584456, 5.37978071273508262860224714024, 5.95312127751398422826120290221, 6.12551683197131201048524442305, 7.16398485061566486905513472860, 7.17809003605299103764493011803, 7.927576317124303856636911842824, 8.925990233820996665478086720144, 9.458264031270819351117623266355, 9.512109378067277170689083727635, 10.50323342657472674293163692824, 10.81827676207360587660495376622, 11.47720137033973757827720181597, 11.73447576609460095821174292080, 12.61593972616659772037447655996

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