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Properties

Label 4-435e2-1.1-c1e2-0-13
Degree $4$
Conductor $189225$
Sign $1$
Analytic cond. $12.0651$
Root an. cond. $1.86373$
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 − 2·3-s − 4-s − 2·5-s + 4·6-s + 8·8-s + 3·9-s + 4·10-s + 2·12-s + 4·15-s − 7·16-s − 12·17-s − 6·18-s + 2·20-s − 16·24-s − 25-s − 4·27-s − 10·29-s − 8·30-s − 14·32-s + 24·34-s − 3·36-s − 4·37-s − 16·40-s + 8·43-s − 6·45-s − 16·47-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s − 1/2·4-s − 0.894·5-s + 1.63·6-s + 2.82·8-s + 9-s + 1.26·10-s + 0.577·12-s + 1.03·15-s − 7/4·16-s − 2.91·17-s − 1.41·18-s + 0.447·20-s − 3.26·24-s − 1/5·25-s − 0.769·27-s − 1.85·29-s − 1.46·30-s − 2.47·32-s + 4.11·34-s − 1/2·36-s − 0.657·37-s − 2.52·40-s + 1.21·43-s − 0.894·45-s − 2.33·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(189225\)    =    \(3^{2} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(12.0651\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 189225,\ (\ :1/2, 1/2),\ 1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad3$C_1$ \( ( 1 + T )^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
29$C_2$ \( 1 + 10 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.31.a_acg
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.47.q_gc
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2^2$ \( 1 - 98 T^{2} + p^{2} T^{4} \) 2.67.a_adu
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - 154 T^{2} + p^{2} T^{4} \) 2.79.a_afy
83$C_2^2$ \( 1 - 162 T^{2} + p^{2} T^{4} \) 2.83.a_agg
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−11.01921948665195736475932590442, −10.52178406902344915203823192479, −9.844541341934799040560767410274, −9.634556378935702051532899832828, −8.962770979758444492105874043015, −8.864268739584839769236239556238, −8.265103481443087573599461206104, −7.80887303549235796345368344227, −7.25922151954115503528134890444, −7.01379600029443245678673128558, −6.32902004363146727163505315588, −5.65141638178028783793222122756, −4.93205745736017389777985557258, −4.68089650527142608611050709132, −3.93255288591268753972748989651, −3.89089687171672985448297674987, −2.22639463145755144244898757008, −1.39884724066138897148454195914, 0, 0, 1.39884724066138897148454195914, 2.22639463145755144244898757008, 3.89089687171672985448297674987, 3.93255288591268753972748989651, 4.68089650527142608611050709132, 4.93205745736017389777985557258, 5.65141638178028783793222122756, 6.32902004363146727163505315588, 7.01379600029443245678673128558, 7.25922151954115503528134890444, 7.80887303549235796345368344227, 8.265103481443087573599461206104, 8.864268739584839769236239556238, 8.962770979758444492105874043015, 9.634556378935702051532899832828, 9.844541341934799040560767410274, 10.52178406902344915203823192479, 11.01921948665195736475932590442

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