Properties

Label 4-8470e2-1.1-c1e2-0-14
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 3·4-s + 2·5-s − 2·6-s − 2·7-s − 4·8-s − 4·9-s − 4·10-s + 3·12-s − 9·13-s + 4·14-s + 2·15-s + 5·16-s − 7·17-s + 8·18-s + 6·19-s + 6·20-s − 2·21-s + 8·23-s − 4·24-s + 3·25-s + 18·26-s − 6·27-s − 6·28-s + 5·29-s − 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s − 0.816·6-s − 0.755·7-s − 1.41·8-s − 4/3·9-s − 1.26·10-s + 0.866·12-s − 2.49·13-s + 1.06·14-s + 0.516·15-s + 5/4·16-s − 1.69·17-s + 1.88·18-s + 1.37·19-s + 1.34·20-s − 0.436·21-s + 1.66·23-s − 0.816·24-s + 3/5·25-s + 3.53·26-s − 1.15·27-s − 1.13·28-s + 0.928·29-s − 0.730·30-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$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 9 T + 45 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 5 T + 53 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 15 T + 119 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 102 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 2 T + 130 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - T + 81 T^{2} - p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 25 T + 321 T^{2} + 25 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 14 T + 182 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 213 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−7.51435016862333867799071201758, −7.39818965370076322839525424556, −6.98899408251584528600183720536, −6.70328894985533100531360673576, −6.36857887485927252639699776255, −6.18252509838450941638277428647, −5.52063715643670033074036937191, −5.21723646533143036010134286238, −4.83139705307050625500049062569, −4.80472356323487439056386726753, −3.81224564744162149699986728843, −3.50139610773278868663914581379, −2.79530560741138540366489672004, −2.67033290476247438678675169710, −2.41156282375535625445918186100, −2.36865826761148818421962458724, −1.30736366624602087019287236653, −1.01410183985042093900036901219, 0, 0, 1.01410183985042093900036901219, 1.30736366624602087019287236653, 2.36865826761148818421962458724, 2.41156282375535625445918186100, 2.67033290476247438678675169710, 2.79530560741138540366489672004, 3.50139610773278868663914581379, 3.81224564744162149699986728843, 4.80472356323487439056386726753, 4.83139705307050625500049062569, 5.21723646533143036010134286238, 5.52063715643670033074036937191, 6.18252509838450941638277428647, 6.36857887485927252639699776255, 6.70328894985533100531360673576, 6.98899408251584528600183720536, 7.39818965370076322839525424556, 7.51435016862333867799071201758

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