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Properties

Label 4-373527-1.1-c1e2-0-7
Degree $4$
Conductor $373527$
Sign $-1$
Analytic cond. $23.8164$
Root an. cond. $2.20911$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·4-s − 7-s + 9-s + 4·11-s − 4·13-s + 5·16-s − 12·17-s + 8·19-s − 6·25-s + 3·28-s − 3·36-s + 12·37-s + 4·41-s − 12·44-s + 49-s + 12·52-s + 12·53-s − 4·61-s − 63-s − 3·64-s + 8·67-s + 36·68-s − 12·73-s − 24·76-s − 4·77-s + 81-s − 24·83-s + ⋯
L(s)  = 1  − 3/2·4-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 5/4·16-s − 2.91·17-s + 1.83·19-s − 6/5·25-s + 0.566·28-s − 1/2·36-s + 1.97·37-s + 0.624·41-s − 1.80·44-s + 1/7·49-s + 1.66·52-s + 1.64·53-s − 0.512·61-s − 0.125·63-s − 3/8·64-s + 0.977·67-s + 4.36·68-s − 1.40·73-s − 2.75·76-s − 0.455·77-s + 1/9·81-s − 2.63·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373527 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(373527\)    =    \(3^{2} \cdot 7^{3} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(23.8164\)
Root analytic conductor: \(2.20911\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 373527,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
7$C_1$ \( 1 + T \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.2.a_d
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.17.m_cs
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.19.ai_cc
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.79.a_adu
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.83.y_ly
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−8.672365196683144779961098572501, −8.116871407954105327881310337349, −7.36940949637622773245609071348, −7.25047783802838427330028450541, −6.62105717552400612885059059406, −6.04825788583317143584553877528, −5.61883729113008416400697263528, −4.88039179056941194097196564281, −4.49790654060497533755621117209, −4.13559084050773741974089362056, −3.73651278750844385913287951941, −2.78382618289105024776834466639, −2.17377383748365885728293696566, −1.05642342442732846317295357874, 0, 1.05642342442732846317295357874, 2.17377383748365885728293696566, 2.78382618289105024776834466639, 3.73651278750844385913287951941, 4.13559084050773741974089362056, 4.49790654060497533755621117209, 4.88039179056941194097196564281, 5.61883729113008416400697263528, 6.04825788583317143584553877528, 6.62105717552400612885059059406, 7.25047783802838427330028450541, 7.36940949637622773245609071348, 8.116871407954105327881310337349, 8.672365196683144779961098572501

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