Properties

Label 4-10110-1.1-c1e2-0-0
Degree $4$
Conductor $10110$
Sign $-1$
Analytic cond. $0.644622$
Root an. cond. $0.896037$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s − 3·5-s − 2·7-s − 2·8-s − 2·9-s + 4·11-s − 6·13-s + 16-s + 3·20-s − 23-s + 2·25-s − 3·27-s + 2·28-s − 9·29-s + 5·31-s + 4·32-s + 6·35-s + 2·36-s − 4·37-s + 6·40-s + 41-s + 43-s − 4·44-s + 6·45-s + 10·47-s + 2·49-s + 6·52-s + ⋯
L(s)  = 1  − 1/2·4-s − 1.34·5-s − 0.755·7-s − 0.707·8-s − 2/3·9-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.670·20-s − 0.208·23-s + 2/5·25-s − 0.577·27-s + 0.377·28-s − 1.67·29-s + 0.898·31-s + 0.707·32-s + 1.01·35-s + 1/3·36-s − 0.657·37-s + 0.948·40-s + 0.156·41-s + 0.152·43-s − 0.603·44-s + 0.894·45-s + 1.45·47-s + 2/7·49-s + 0.832·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10110 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10110 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10110\)    =    \(2 \cdot 3 \cdot 5 \cdot 337\)
Sign: $-1$
Analytic conductor: \(0.644622\)
Root analytic conductor: \(0.896037\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10110,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
3$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - T + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
337$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 14 T + p T^{2} ) \)
good7$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + T - 18 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 5 T + 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T - 38 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
53$D_{4}$ \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 15 T + 132 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 114 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
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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

−16.8473292947, −16.4556184439, −15.6807975858, −15.2836989777, −14.9234201756, −14.5046209034, −13.8691649464, −13.4381525226, −12.5097568494, −12.2257064786, −11.9638837862, −11.4015555406, −10.8155216924, −9.86726009028, −9.52308285391, −9.01682856506, −8.46566722300, −7.63303028956, −7.32255497198, −6.45781626909, −5.82157369982, −4.98558127776, −4.03601459424, −3.62475947716, −2.61548592535, 0, 2.61548592535, 3.62475947716, 4.03601459424, 4.98558127776, 5.82157369982, 6.45781626909, 7.32255497198, 7.63303028956, 8.46566722300, 9.01682856506, 9.52308285391, 9.86726009028, 10.8155216924, 11.4015555406, 11.9638837862, 12.2257064786, 12.5097568494, 13.4381525226, 13.8691649464, 14.5046209034, 14.9234201756, 15.2836989777, 15.6807975858, 16.4556184439, 16.8473292947

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