Properties

Label 4-8470e2-1.1-c1e2-0-15
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 + 3·3-s + 3·4-s − 2·5-s − 6·6-s − 2·7-s − 4·8-s + 2·9-s + 4·10-s + 9·12-s − 4·13-s + 4·14-s − 6·15-s + 5·16-s + 17-s − 4·18-s − 7·19-s − 6·20-s − 6·21-s + 12·23-s − 12·24-s + 3·25-s + 8·26-s − 6·27-s − 6·28-s − 2·29-s + 12·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.73·3-s + 3/2·4-s − 0.894·5-s − 2.44·6-s − 0.755·7-s − 1.41·8-s + 2/3·9-s + 1.26·10-s + 2.59·12-s − 1.10·13-s + 1.06·14-s − 1.54·15-s + 5/4·16-s + 0.242·17-s − 0.942·18-s − 1.60·19-s − 1.34·20-s − 1.30·21-s + 2.50·23-s − 2.44·24-s + 3/5·25-s + 1.56·26-s − 1.15·27-s − 1.13·28-s − 0.371·29-s + 2.19·30-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$D_{4}$ \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$D_{4}$ \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_4$ \( 1 + 7 T + 39 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 3 T + 77 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 13 T + 159 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
61$C_4$ \( 1 - 16 T + 166 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 103 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 106 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 15 T + 203 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - T + 93 T^{2} - p T^{3} + p^{2} T^{4} \)
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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

−7.57880156092898263352805140912, −7.46264429169674004532743692244, −7.21189559697520536471377097697, −6.87594465866567570213894041460, −6.34922902067241816381709942199, −6.25558135739230496028405748544, −5.42167959254139053064542883107, −5.38928741433899459529737440625, −4.67816911348486045534147491870, −4.30819419361542607104746686627, −3.71539330290363594427277486396, −3.66185719641731472701115974086, −2.90981253364544447084092289390, −2.84849362557972434686489115053, −2.38245051820441006457868111340, −2.34397077772204736041348114058, −1.37022022837728817710405307204, −1.02426882443860840444541055592, 0, 0, 1.02426882443860840444541055592, 1.37022022837728817710405307204, 2.34397077772204736041348114058, 2.38245051820441006457868111340, 2.84849362557972434686489115053, 2.90981253364544447084092289390, 3.66185719641731472701115974086, 3.71539330290363594427277486396, 4.30819419361542607104746686627, 4.67816911348486045534147491870, 5.38928741433899459529737440625, 5.42167959254139053064542883107, 6.25558135739230496028405748544, 6.34922902067241816381709942199, 6.87594465866567570213894041460, 7.21189559697520536471377097697, 7.46264429169674004532743692244, 7.57880156092898263352805140912

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