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Properties

Label 4-930e2-1.1-c1e2-0-11
Degree $4$
Conductor $864900$
Sign $1$
Analytic cond. $55.1467$
Root an. cond. $2.72508$
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  − 4-s + 4·5-s − 9-s − 4·11-s + 16-s + 16·19-s − 4·20-s + 11·25-s − 8·29-s + 2·31-s + 36-s − 12·41-s + 4·44-s − 4·45-s + 10·49-s − 16·55-s + 28·59-s − 4·61-s − 64-s + 24·71-s − 16·76-s + 32·79-s + 4·80-s + 81-s − 12·89-s + 64·95-s + 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s − 1.20·11-s + 1/4·16-s + 3.67·19-s − 0.894·20-s + 11/5·25-s − 1.48·29-s + 0.359·31-s + 1/6·36-s − 1.87·41-s + 0.603·44-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 3.64·59-s − 0.512·61-s − 1/8·64-s + 2.84·71-s − 1.83·76-s + 3.60·79-s + 0.447·80-s + 1/9·81-s − 1.27·89-s + 6.56·95-s + 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(864900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(55.1467\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 864900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.636523682\)
\(L(\frac12)\) \(\approx\) \(2.636523682\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
31$C_1$ \( ( 1 - T )^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.19.aq_dy
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.59.abc_mc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.73.a_afq
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−9.992703882407293044710947233183, −9.834232487658171719532765214120, −9.520603337532747126115513738503, −9.258641200621032622349906959493, −8.727133041099525573745914304987, −8.034241419903732867579777149116, −7.975424352582576624268774178038, −7.22897813146381006534951537156, −6.90816978743354370729078188135, −6.43261538440133354631313820935, −5.51534312916078272336124634205, −5.47584513291779601327362567906, −5.27388207243613997790328100551, −4.95701155419698269502046728540, −3.68981082289944354077731147925, −3.57913206698963702191270578467, −2.60679097366355926524178899677, −2.48263301015210684269712312384, −1.50113647328993692059699899371, −0.827578532815388943043566540273, 0.827578532815388943043566540273, 1.50113647328993692059699899371, 2.48263301015210684269712312384, 2.60679097366355926524178899677, 3.57913206698963702191270578467, 3.68981082289944354077731147925, 4.95701155419698269502046728540, 5.27388207243613997790328100551, 5.47584513291779601327362567906, 5.51534312916078272336124634205, 6.43261538440133354631313820935, 6.90816978743354370729078188135, 7.22897813146381006534951537156, 7.975424352582576624268774178038, 8.034241419903732867579777149116, 8.727133041099525573745914304987, 9.258641200621032622349906959493, 9.520603337532747126115513738503, 9.834232487658171719532765214120, 9.992703882407293044710947233183

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