Properties

Label 4-2016e2-1.1-c1e2-0-13
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  − 5·7-s − 2·11-s + 2·13-s − 2·17-s − 5·19-s + 6·23-s + 5·25-s + 16·29-s + 3·31-s + 9·37-s − 4·41-s + 2·43-s + 8·47-s + 18·49-s + 6·53-s + 6·59-s + 2·61-s + 5·67-s − 8·71-s + 11·73-s + 10·77-s + 5·79-s + 12·89-s − 10·91-s + 36·97-s − 6·101-s + 11·103-s + ⋯
L(s)  = 1  − 1.88·7-s − 0.603·11-s + 0.554·13-s − 0.485·17-s − 1.14·19-s + 1.25·23-s + 25-s + 2.97·29-s + 0.538·31-s + 1.47·37-s − 0.624·41-s + 0.304·43-s + 1.16·47-s + 18/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s + 0.610·67-s − 0.949·71-s + 1.28·73-s + 1.13·77-s + 0.562·79-s + 1.27·89-s − 1.04·91-s + 3.65·97-s − 0.597·101-s + 1.08·103-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

\(L(1)\) \(\approx\) \(2.106699033\)
\(L(\frac12)\) \(\approx\) \(2.106699033\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
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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

−9.210220126892870990936180394985, −8.802887512635859271236806134655, −8.729983498780691958254515901069, −8.428174046964265616624622357564, −7.57565832573131320214356719704, −7.51874296049311745459624114336, −6.67866005004160149304255744705, −6.63368033627424095526584596572, −6.29429945650867780502041776543, −6.04700827487207487213898640207, −5.20647901203443763894440876646, −5.01781905411925733435518817575, −4.30573812498332139594262724586, −4.14319712929519341650642299714, −3.23645231162302387639856389423, −3.18319680120722497899312602920, −2.39035303845600298420578278056, −2.38941155371295873847834442547, −0.835380808261731764761490791627, −0.75548003203376718681735610263, 0.75548003203376718681735610263, 0.835380808261731764761490791627, 2.38941155371295873847834442547, 2.39035303845600298420578278056, 3.18319680120722497899312602920, 3.23645231162302387639856389423, 4.14319712929519341650642299714, 4.30573812498332139594262724586, 5.01781905411925733435518817575, 5.20647901203443763894440876646, 6.04700827487207487213898640207, 6.29429945650867780502041776543, 6.63368033627424095526584596572, 6.67866005004160149304255744705, 7.51874296049311745459624114336, 7.57565832573131320214356719704, 8.428174046964265616624622357564, 8.729983498780691958254515901069, 8.802887512635859271236806134655, 9.210220126892870990936180394985

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