Properties

Label 4-2016e2-1.1-c1e2-0-7
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

\(L(1)\) \(\approx\) \(3.234569489\)
\(L(\frac12)\) \(\approx\) \(3.234569489\)
\(L(\frac{3}{2})\) 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

−9.525774627127339485037275891585, −8.736691780547568924835719545975, −8.685204516555070348500925097758, −8.346747057119739796582791129875, −7.65793701233622131700668379039, −7.38534481682280194667432039612, −7.20361048881376163126012316129, −6.43638663487877139478220909891, −6.07935480851325592963968882750, −5.67423549159683224908794303725, −5.46896932051028645702667004199, −5.03700057765526876492813317456, −4.66948933101685461394439327788, −3.93798226426328505923379919355, −3.40943219604235664577552860876, −2.99938453640640460508747082895, −2.50033588674012115380288386634, −1.80777817784617706108728586105, −1.51261375670747518773645512088, −0.64836441377961573383700110765, 0.64836441377961573383700110765, 1.51261375670747518773645512088, 1.80777817784617706108728586105, 2.50033588674012115380288386634, 2.99938453640640460508747082895, 3.40943219604235664577552860876, 3.93798226426328505923379919355, 4.66948933101685461394439327788, 5.03700057765526876492813317456, 5.46896932051028645702667004199, 5.67423549159683224908794303725, 6.07935480851325592963968882750, 6.43638663487877139478220909891, 7.20361048881376163126012316129, 7.38534481682280194667432039612, 7.65793701233622131700668379039, 8.346747057119739796582791129875, 8.685204516555070348500925097758, 8.736691780547568924835719545975, 9.525774627127339485037275891585

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