Properties

Label 4-8470e2-1.1-c1e2-0-9
Degree $4$
Conductor $71740900$
Sign $1$
Analytic cond. $4574.26$
Root an. cond. $8.22394$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 2·5-s − 2·7-s − 4·8-s − 3·9-s + 4·10-s + 4·14-s + 5·16-s + 6·17-s + 6·18-s − 6·20-s − 2·23-s + 3·25-s − 6·28-s + 12·29-s − 10·31-s − 6·32-s − 12·34-s + 4·35-s − 9·36-s − 8·37-s + 8·40-s − 10·41-s + 2·43-s + 6·45-s + 4·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s − 1.41·8-s − 9-s + 1.26·10-s + 1.06·14-s + 5/4·16-s + 1.45·17-s + 1.41·18-s − 1.34·20-s − 0.417·23-s + 3/5·25-s − 1.13·28-s + 2.22·29-s − 1.79·31-s − 1.06·32-s − 2.05·34-s + 0.676·35-s − 3/2·36-s − 1.31·37-s + 1.26·40-s − 1.56·41-s + 0.304·43-s + 0.894·45-s + 0.589·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71740900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(71740900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(4574.26\)
Root analytic conductor: \(8.22394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8470} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 71740900,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 10 T + 104 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 6 T + 112 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 - 18 T + 188 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 159 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 10 T + 143 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 184 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 4 T + 186 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
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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

−7.50739695353278018964179195550, −7.34137835118480474628921054727, −7.19388363328662825066792092611, −6.67803975695054432128094851055, −6.31257310012961913228278591399, −6.13092994185035302680127168585, −5.53208802432604453052306188894, −5.36769663489001371646817226634, −4.91634097767926948483642862377, −4.46089811064048702901916626141, −3.72460042336042834214211724687, −3.57134596045895562263583275975, −3.14734619519579952373366198937, −3.01010737437219644387984894911, −2.31259869164806212995133579681, −1.98259430875505756494115095322, −1.23608780910489650602327876673, −0.894777808514120225846658855306, 0, 0, 0.894777808514120225846658855306, 1.23608780910489650602327876673, 1.98259430875505756494115095322, 2.31259869164806212995133579681, 3.01010737437219644387984894911, 3.14734619519579952373366198937, 3.57134596045895562263583275975, 3.72460042336042834214211724687, 4.46089811064048702901916626141, 4.91634097767926948483642862377, 5.36769663489001371646817226634, 5.53208802432604453052306188894, 6.13092994185035302680127168585, 6.31257310012961913228278591399, 6.67803975695054432128094851055, 7.19388363328662825066792092611, 7.34137835118480474628921054727, 7.50739695353278018964179195550

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