Properties

Label 4-112e2-1.1-c1e2-0-13
Degree $4$
Conductor $12544$
Sign $1$
Analytic cond. $0.799816$
Root an. cond. $0.945687$
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  + 4·3-s − 4·7-s + 6·9-s + 4·19-s − 16·21-s − 2·25-s − 4·27-s + 12·29-s − 16·31-s − 4·37-s + 9·49-s + 12·53-s + 16·57-s + 12·59-s − 24·63-s − 8·75-s − 37·81-s − 12·83-s + 48·87-s − 64·93-s + 8·103-s + 28·109-s − 16·111-s − 36·113-s + 10·121-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2.30·3-s − 1.51·7-s + 2·9-s + 0.917·19-s − 3.49·21-s − 2/5·25-s − 0.769·27-s + 2.22·29-s − 2.87·31-s − 0.657·37-s + 9/7·49-s + 1.64·53-s + 2.11·57-s + 1.56·59-s − 3.02·63-s − 0.923·75-s − 4.11·81-s − 1.31·83-s + 5.14·87-s − 6.63·93-s + 0.788·103-s + 2.68·109-s − 1.51·111-s − 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12544\)    =    \(2^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.799816\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12544,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724747304\)
\(L(\frac12)\) \(\approx\) \(1.724747304\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 146 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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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

−13.85527597205343874413781727946, −13.51800823632316294200121117071, −13.08341381244233473952115796668, −12.59967805565372363074940228804, −11.95841262022681953421076004057, −11.34442682559592549094610972239, −10.45584994758087072117626657560, −9.921506810920154637386781506468, −9.507527997314415692233235404029, −9.003599649400614938492986071576, −8.614121972701083607506245823139, −8.128628623235270634301562672712, −7.20708683661451296405070797755, −7.11598076117609151613707720329, −6.00442836075500351053765247795, −5.36827972142785832925545706293, −3.99977327571498627519619112045, −3.45534973712552675432088105877, −2.96568982120453400149957970307, −2.17971886719975111308459041619, 2.17971886719975111308459041619, 2.96568982120453400149957970307, 3.45534973712552675432088105877, 3.99977327571498627519619112045, 5.36827972142785832925545706293, 6.00442836075500351053765247795, 7.11598076117609151613707720329, 7.20708683661451296405070797755, 8.128628623235270634301562672712, 8.614121972701083607506245823139, 9.003599649400614938492986071576, 9.507527997314415692233235404029, 9.921506810920154637386781506468, 10.45584994758087072117626657560, 11.34442682559592549094610972239, 11.95841262022681953421076004057, 12.59967805565372363074940228804, 13.08341381244233473952115796668, 13.51800823632316294200121117071, 13.85527597205343874413781727946

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