Properties

Label 4-114e2-1.1-c1e2-0-7
Degree $4$
Conductor $12996$
Sign $1$
Analytic cond. $0.828636$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 6-s + 2·7-s − 8-s − 4·11-s + 3·13-s + 2·14-s − 16-s − 4·17-s + 8·19-s + 2·21-s − 4·22-s − 4·23-s − 24-s + 5·25-s + 3·26-s − 27-s − 6·31-s − 4·33-s − 4·34-s − 10·37-s + 8·38-s + 3·39-s − 4·41-s + 2·42-s + 9·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.408·6-s + 0.755·7-s − 0.353·8-s − 1.20·11-s + 0.832·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 1.83·19-s + 0.436·21-s − 0.852·22-s − 0.834·23-s − 0.204·24-s + 25-s + 0.588·26-s − 0.192·27-s − 1.07·31-s − 0.696·33-s − 0.685·34-s − 1.64·37-s + 1.29·38-s + 0.480·39-s − 0.624·41-s + 0.308·42-s + 1.37·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12996\)    =    \(2^{2} \cdot 3^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(0.828636\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{114} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675438852\)
\(L(\frac12)\) \(\approx\) \(1.675438852\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
3$C_2$ \( 1 - T + T^{2} \)
19$C_2$ \( 1 - 8 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 4 T - 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 14 T + 125 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.88935122386104145733075453711, −13.33372181003026943671910859257, −13.09184942614252579263172866433, −12.42716835803818488674609293872, −11.75746320541781387013962392169, −11.41664737948096430275132660373, −10.63941498532250438456701673974, −10.44669732874482476927455045214, −9.403497119786081352456293968974, −9.113955766948714279850299130543, −8.280834324274769893054065768919, −8.042395623129522344473548287943, −7.28796894625811906590406858420, −6.63182940809195906115128037212, −5.68181664238609595037831782148, −5.19544346410358851280221525081, −4.63444084371942773196590408353, −3.61870851811828748658304663038, −3.03965187205450842041976789976, −1.88720241946061314769023443231, 1.88720241946061314769023443231, 3.03965187205450842041976789976, 3.61870851811828748658304663038, 4.63444084371942773196590408353, 5.19544346410358851280221525081, 5.68181664238609595037831782148, 6.63182940809195906115128037212, 7.28796894625811906590406858420, 8.042395623129522344473548287943, 8.280834324274769893054065768919, 9.113955766948714279850299130543, 9.403497119786081352456293968974, 10.44669732874482476927455045214, 10.63941498532250438456701673974, 11.41664737948096430275132660373, 11.75746320541781387013962392169, 12.42716835803818488674609293872, 13.09184942614252579263172866433, 13.33372181003026943671910859257, 13.88935122386104145733075453711

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