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Properties

Label 4-273e2-1.1-c1e2-0-21
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 $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·7-s − 3·9-s − 2·13-s − 4·16-s + 10·19-s − 25-s − 6·31-s − 8·37-s − 2·43-s + 3·49-s − 20·61-s + 6·63-s − 12·67-s − 26·73-s + 6·79-s + 9·81-s + 4·91-s + 14·97-s − 8·103-s − 4·109-s + 8·112-s + 6·117-s + 14·121-s + 127-s + 131-s − 20·133-s + 137-s + ⋯
L(s)  = 1  − 0.755·7-s − 9-s − 0.554·13-s − 16-s + 2.29·19-s − 1/5·25-s − 1.07·31-s − 1.31·37-s − 0.304·43-s + 3/7·49-s − 2.56·61-s + 0.755·63-s − 1.46·67-s − 3.04·73-s + 0.675·79-s + 81-s + 0.419·91-s + 1.42·97-s − 0.788·103-s − 0.383·109-s + 0.755·112-s + 0.554·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s − 1.73·133-s + 0.0854·137-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: \(1\)
Selberg data: \((4,\ 74529,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.19.ak_cl
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.23.a_bl
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.a_bh
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.31.g_ct
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.a_z
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.67.m_go
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.73.ba_md
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.79.ag_gl
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−9.362259319494049279221007732110, −8.998346772410808154035420365472, −8.895694395064855641013305453407, −7.88305646625584781075065151406, −7.43563929744309328324545511818, −7.14095628036298532453823579927, −6.35409767437233023841602048217, −5.88871707105200974021405692252, −5.27883313770925671699270459709, −4.85246247359459828104814508190, −3.93866594968967426469354468549, −3.11744276397134147631028514628, −2.87743708330085578837178550046, −1.68695310650066648413239890440, 0, 1.68695310650066648413239890440, 2.87743708330085578837178550046, 3.11744276397134147631028514628, 3.93866594968967426469354468549, 4.85246247359459828104814508190, 5.27883313770925671699270459709, 5.88871707105200974021405692252, 6.35409767437233023841602048217, 7.14095628036298532453823579927, 7.43563929744309328324545511818, 7.88305646625584781075065151406, 8.895694395064855641013305453407, 8.998346772410808154035420365472, 9.362259319494049279221007732110

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