Properties

Label 4-2156-1.1-c1e2-0-2
Degree $4$
Conductor $2156$
Sign $1$
Analytic cond. $0.137468$
Root an. cond. $0.608906$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4-s − 4·7-s − 2·9-s − 11-s + 16-s + 2·25-s − 4·28-s + 12·29-s − 2·36-s − 8·37-s − 8·43-s − 44-s + 9·49-s + 8·63-s + 64-s + 4·67-s + 12·71-s + 4·77-s − 8·79-s − 5·81-s + 2·99-s + 2·100-s − 12·107-s + 16·109-s − 4·112-s + 12·113-s + 12·116-s + ⋯
L(s)  = 1  + 1/2·4-s − 1.51·7-s − 2/3·9-s − 0.301·11-s + 1/4·16-s + 2/5·25-s − 0.755·28-s + 2.22·29-s − 1/3·36-s − 1.31·37-s − 1.21·43-s − 0.150·44-s + 9/7·49-s + 1.00·63-s + 1/8·64-s + 0.488·67-s + 1.42·71-s + 0.455·77-s − 0.900·79-s − 5/9·81-s + 0.201·99-s + 1/5·100-s − 1.16·107-s + 1.53·109-s − 0.377·112-s + 1.12·113-s + 1.11·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2156\)    =    \(2^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.137468\)
Root analytic conductor: \(0.608906\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2156} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2156,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6353160322\)
\(L(\frac12)\) \(\approx\) \(0.6353160322\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - T )( 1 + T ) \)
7$C_2$ \( 1 + 4 T + p T^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
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

−13.14444196011017736624954771284, −12.58572238817061641688555604504, −12.07640046661767485447135762696, −11.48746584495738347692356080454, −10.61647521625764182154291152125, −10.18774784796246493803027109873, −9.571837480671157198635959777949, −8.691923532445567953257811423624, −8.199522970356618712223090269825, −7.04378127151225446544122784756, −6.61188575429027541858457662119, −5.88669620904765581852775751295, −4.91759000072777764587960332982, −3.49031776546888561563567080235, −2.71835047120874446997595797974, 2.71835047120874446997595797974, 3.49031776546888561563567080235, 4.91759000072777764587960332982, 5.88669620904765581852775751295, 6.61188575429027541858457662119, 7.04378127151225446544122784756, 8.199522970356618712223090269825, 8.691923532445567953257811423624, 9.571837480671157198635959777949, 10.18774784796246493803027109873, 10.61647521625764182154291152125, 11.48746584495738347692356080454, 12.07640046661767485447135762696, 12.58572238817061641688555604504, 13.14444196011017736624954771284

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