Properties

Label 4-56e2-1.1-c1e2-0-2
Degree $4$
Conductor $3136$
Sign $1$
Analytic cond. $0.199954$
Root an. cond. $0.668701$
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  − 2·4-s + 2·7-s + 4·9-s + 4·16-s − 12·17-s − 12·23-s + 8·25-s − 4·28-s − 8·31-s − 8·36-s + 12·41-s + 3·49-s + 8·63-s − 8·64-s + 24·68-s + 4·73-s + 16·79-s + 7·81-s + 12·89-s + 24·92-s − 20·97-s − 16·100-s − 8·103-s + 8·112-s − 24·113-s − 24·119-s + 14·121-s + ⋯
L(s)  = 1  − 4-s + 0.755·7-s + 4/3·9-s + 16-s − 2.91·17-s − 2.50·23-s + 8/5·25-s − 0.755·28-s − 1.43·31-s − 4/3·36-s + 1.87·41-s + 3/7·49-s + 1.00·63-s − 64-s + 2.91·68-s + 0.468·73-s + 1.80·79-s + 7/9·81-s + 1.27·89-s + 2.50·92-s − 2.03·97-s − 8/5·100-s − 0.788·103-s + 0.755·112-s − 2.25·113-s − 2.20·119-s + 1.27·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.199954\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3136,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6681890356\)
\(L(\frac12)\) \(\approx\) \(0.6681890356\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 76 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−15.49930421278808901056139214054, −14.96452140700649237510354520469, −14.45766830375256182878283408496, −13.76257516078855623119689374270, −13.44118598925179590146497719669, −12.66468291984322253013169299650, −12.53347172894226953103425392872, −11.55515775923201473995815722510, −10.69373036516441500804947735799, −10.62691907987965616656245148422, −9.492639763691787647890359277125, −9.210651331591415488337301882557, −8.442592762612388001199889949344, −7.84323095250001045709995789706, −7.02844486450375570678104241254, −6.28897480723706332234728122835, −5.20312920827363574835035017621, −4.29346346328765948086659099522, −4.13452873933357371408227588941, −2.05976071722565345195527791409, 2.05976071722565345195527791409, 4.13452873933357371408227588941, 4.29346346328765948086659099522, 5.20312920827363574835035017621, 6.28897480723706332234728122835, 7.02844486450375570678104241254, 7.84323095250001045709995789706, 8.442592762612388001199889949344, 9.210651331591415488337301882557, 9.492639763691787647890359277125, 10.62691907987965616656245148422, 10.69373036516441500804947735799, 11.55515775923201473995815722510, 12.53347172894226953103425392872, 12.66468291984322253013169299650, 13.44118598925179590146497719669, 13.76257516078855623119689374270, 14.45766830375256182878283408496, 14.96452140700649237510354520469, 15.49930421278808901056139214054

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