Properties

Label 4-2016e2-1.1-c1e2-0-10
Degree $4$
Conductor $4064256$
Sign $1$
Analytic cond. $259.140$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s − 7-s + 5·11-s + 4·13-s + 2·17-s − 6·19-s + 2·23-s + 5·25-s − 2·29-s − 31-s − 3·35-s − 10·37-s − 8·41-s − 8·43-s − 8·47-s − 6·49-s + 5·53-s + 15·55-s + 13·59-s + 8·61-s + 12·65-s − 14·67-s + 24·71-s + 6·73-s − 5·77-s + 11·79-s + 14·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 0.377·7-s + 1.50·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s + 0.417·23-s + 25-s − 0.371·29-s − 0.179·31-s − 0.507·35-s − 1.64·37-s − 1.24·41-s − 1.21·43-s − 1.16·47-s − 6/7·49-s + 0.686·53-s + 2.02·55-s + 1.69·59-s + 1.02·61-s + 1.48·65-s − 1.71·67-s + 2.84·71-s + 0.702·73-s − 0.569·77-s + 1.23·79-s + 1.53·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4064256\)    =    \(2^{10} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(259.140\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2016} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4064256,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.559369001\)
\(L(\frac12)\) \(\approx\) \(3.559369001\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_2$ \( 1 + T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 5 T - 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )^{2} \)
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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

−9.235369252272996513624492306330, −9.045123125375369896602600424744, −8.602747297073097297863471648108, −8.353570762400622730150495332305, −7.971922064393967368105895191161, −7.13101119117500357847415777651, −6.76616668862170428917346915030, −6.65948385872621481819399551257, −6.09894701559066043476933716329, −6.06752177642340291844027681224, −5.22780512147982155854766031564, −5.10967807210319452549296861953, −4.54431664561376854670460912610, −3.74719883940556922878294168727, −3.50294567148059797261250200292, −3.32034840118606920397202253026, −2.12160047436661961584688419465, −2.03755895101904677500674640748, −1.44327569818942655727504395071, −0.69586354198587284954155924303, 0.69586354198587284954155924303, 1.44327569818942655727504395071, 2.03755895101904677500674640748, 2.12160047436661961584688419465, 3.32034840118606920397202253026, 3.50294567148059797261250200292, 3.74719883940556922878294168727, 4.54431664561376854670460912610, 5.10967807210319452549296861953, 5.22780512147982155854766031564, 6.06752177642340291844027681224, 6.09894701559066043476933716329, 6.65948385872621481819399551257, 6.76616668862170428917346915030, 7.13101119117500357847415777651, 7.971922064393967368105895191161, 8.353570762400622730150495332305, 8.602747297073097297863471648108, 9.045123125375369896602600424744, 9.235369252272996513624492306330

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