Properties

Label 4-273e2-1.1-c1e2-0-31
Degree $4$
Conductor $74529$
Sign $1$
Analytic cond. $4.75203$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·2-s + 3-s + 4·4-s + 6·5-s + 3·6-s − 7-s + 3·8-s + 18·10-s − 9·11-s + 4·12-s − 2·13-s − 3·14-s + 6·15-s + 3·16-s + 3·17-s − 9·19-s + 24·20-s − 21-s − 27·22-s − 6·23-s + 3·24-s + 19·25-s − 6·26-s − 27-s − 4·28-s + 6·29-s + 18·30-s + ⋯
L(s)  = 1  + 2.12·2-s + 0.577·3-s + 2·4-s + 2.68·5-s + 1.22·6-s − 0.377·7-s + 1.06·8-s + 5.69·10-s − 2.71·11-s + 1.15·12-s − 0.554·13-s − 0.801·14-s + 1.54·15-s + 3/4·16-s + 0.727·17-s − 2.06·19-s + 5.36·20-s − 0.218·21-s − 5.75·22-s − 1.25·23-s + 0.612·24-s + 19/5·25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + 1.11·29-s + 3.28·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(74529\)    =    \(3^{2} \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(4.75203\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.125011457\)
\(L(\frac12)\) \(\approx\) \(6.125011457\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 + T + p T^{2} \)
13$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2^2$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
5$C_2^2$ \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T + 49 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 15 T + 134 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2^2$ \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 18 T + 181 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 6 T + 101 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 106 T^{2} + p^{2} T^{4} \)
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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

−12.70187410241916295868444410456, −12.24890137542472350220866597875, −11.07190447606080138406345064927, −10.62086334778531193934628477378, −10.31803819795874944537358884081, −9.877489647000114415312745183109, −9.573669773150660070830699850600, −8.848045810380763734719560423030, −8.117348889148761797760378546942, −7.80493474134101319127737302664, −6.92344577144066472899688655164, −6.19057013816644608343017777255, −5.83359462454362716338919437435, −5.59948891776566973267832108567, −5.07036367408798051558518761101, −4.57101742365766391650417719538, −3.79705292345275806543833505104, −2.74489311513659311615471856407, −2.50665517112265774253943274515, −2.07685404008528896978942627280, 2.07685404008528896978942627280, 2.50665517112265774253943274515, 2.74489311513659311615471856407, 3.79705292345275806543833505104, 4.57101742365766391650417719538, 5.07036367408798051558518761101, 5.59948891776566973267832108567, 5.83359462454362716338919437435, 6.19057013816644608343017777255, 6.92344577144066472899688655164, 7.80493474134101319127737302664, 8.117348889148761797760378546942, 8.848045810380763734719560423030, 9.573669773150660070830699850600, 9.877489647000114415312745183109, 10.31803819795874944537358884081, 10.62086334778531193934628477378, 11.07190447606080138406345064927, 12.24890137542472350220866597875, 12.70187410241916295868444410456

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