Properties

Label 4-8470e2-1.1-c1e2-0-8
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 + 8·13-s + 4·14-s + 5·16-s − 2·17-s + 6·18-s + 4·19-s − 6·20-s − 10·23-s + 3·25-s − 16·26-s − 6·28-s − 16·29-s − 2·31-s − 6·32-s + 4·34-s + 4·35-s − 9·36-s + 12·37-s − 8·38-s + 8·40-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 + 2.21·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s + 1.41·18-s + 0.917·19-s − 1.34·20-s − 2.08·23-s + 3/5·25-s − 3.13·26-s − 1.13·28-s − 2.97·29-s − 0.359·31-s − 1.06·32-s + 0.685·34-s + 0.676·35-s − 3/2·36-s + 1.97·37-s − 1.29·38-s + 1.26·40-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: Trivial
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$D_{4}$ \( 1 - 8 T + 3 p T^{2} - 8 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 39 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 80 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 18 T + 187 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 94 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 6 T + 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 10 T + 128 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.79808346476681715349104937063, −7.54066265604383766733640420031, −7.00116879226255352271909234808, −6.70053608739809852477112962485, −6.09313845145714771319559003235, −6.02455824115979350411303974925, −5.74643796539648033216495359455, −5.64434103993468672148402415541, −4.71321165744082534783722713024, −4.28446699869775921668057473024, −3.85556968358981354744066863022, −3.66330017643938442737818314132, −3.09196599858977200362851897482, −3.06243181597717704350034581675, −2.18913307032200890692767254056, −2.01281206688825571003090974479, −1.29438442649993723000930310360, −0.881100865411984934444298050466, 0, 0, 0.881100865411984934444298050466, 1.29438442649993723000930310360, 2.01281206688825571003090974479, 2.18913307032200890692767254056, 3.06243181597717704350034581675, 3.09196599858977200362851897482, 3.66330017643938442737818314132, 3.85556968358981354744066863022, 4.28446699869775921668057473024, 4.71321165744082534783722713024, 5.64434103993468672148402415541, 5.74643796539648033216495359455, 6.02455824115979350411303974925, 6.09313845145714771319559003235, 6.70053608739809852477112962485, 7.00116879226255352271909234808, 7.54066265604383766733640420031, 7.79808346476681715349104937063

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