Properties

Label 4-114e2-1.1-c1e2-0-4
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  − 3-s + 4-s + 6·7-s − 2·9-s − 12-s − 2·13-s + 16-s − 2·19-s − 6·21-s + 6·25-s + 5·27-s + 6·28-s − 16·31-s − 2·36-s − 4·37-s + 2·39-s + 8·43-s − 48-s + 13·49-s − 2·52-s + 2·57-s + 4·61-s − 12·63-s + 64-s + 6·67-s + 18·73-s − 6·75-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/2·4-s + 2.26·7-s − 2/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.458·19-s − 1.30·21-s + 6/5·25-s + 0.962·27-s + 1.13·28-s − 2.87·31-s − 1/3·36-s − 0.657·37-s + 0.320·39-s + 1.21·43-s − 0.144·48-s + 13/7·49-s − 0.277·52-s + 0.264·57-s + 0.512·61-s − 1.51·63-s + 1/8·64-s + 0.733·67-s + 2.10·73-s − 0.692·75-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12996,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.114350639\)
\(L(\frac12)\) \(\approx\) \(1.114350639\)
\(L(\frac{3}{2})\) 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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_2$ \( 1 + T + p T^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
23$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
59$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−11.07673504239234599196375456535, −10.98850264170077345648603277341, −10.60506565264478658567127520158, −9.607471665269029768409922323294, −8.900666531045489798996891080014, −8.408486909011940238981299566497, −7.892225159593170281658461026008, −7.25038064772684706053554512263, −6.73926084905849932341893089023, −5.70493789281341106083458944039, −5.28069074518443602737935578596, −4.80751556264464963514931464778, −3.86057015395193111701613139713, −2.55734550860524537659292391699, −1.62231881828892009952147969605, 1.62231881828892009952147969605, 2.55734550860524537659292391699, 3.86057015395193111701613139713, 4.80751556264464963514931464778, 5.28069074518443602737935578596, 5.70493789281341106083458944039, 6.73926084905849932341893089023, 7.25038064772684706053554512263, 7.892225159593170281658461026008, 8.408486909011940238981299566497, 8.900666531045489798996891080014, 9.607471665269029768409922323294, 10.60506565264478658567127520158, 10.98850264170077345648603277341, 11.07673504239234599196375456535

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