Properties

Label 4-27040-1.1-c1e2-0-3
Degree $4$
Conductor $27040$
Sign $-1$
Analytic cond. $1.72409$
Root an. cond. $1.14588$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s − 4·9-s + 10-s − 4·13-s + 16-s + 4·18-s − 20-s − 2·25-s + 4·26-s − 2·29-s − 32-s − 4·36-s + 40-s − 12·41-s + 4·45-s − 10·49-s + 2·50-s − 4·52-s + 2·53-s + 2·58-s − 10·61-s + 64-s + 4·65-s + 4·72-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 4/3·9-s + 0.316·10-s − 1.10·13-s + 1/4·16-s + 0.942·18-s − 0.223·20-s − 2/5·25-s + 0.784·26-s − 0.371·29-s − 0.176·32-s − 2/3·36-s + 0.158·40-s − 1.87·41-s + 0.596·45-s − 1.42·49-s + 0.282·50-s − 0.554·52-s + 0.274·53-s + 0.262·58-s − 1.28·61-s + 1/8·64-s + 0.496·65-s + 0.471·72-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040\)    =    \(2^{5} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(1.72409\)
Root analytic conductor: \(1.14588\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 27040,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.19.a_ak
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.c_bi
31$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.31.a_u
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.m_dy
43$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.43.a_q
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ac_de
59$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.59.a_cs
61$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.k_es
67$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.67.a_by
71$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.71.a_abo
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.89.aq_je
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.k_by
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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

−10.26906316462456696937320413075, −9.797102992710542232518309984184, −9.286990966849537833715460133347, −8.605693869432553999354344351766, −8.376385804842880572823578036385, −7.59918729163757270369104716022, −7.32826615231018273816399325592, −6.48484357198194845975797217823, −5.96543580075354110580505048215, −5.21561954477899575985853167184, −4.63563640087801479224504934912, −3.51379424078161490339397601364, −2.91997526003955477197730945842, −1.94404617929522536547252822356, 0, 1.94404617929522536547252822356, 2.91997526003955477197730945842, 3.51379424078161490339397601364, 4.63563640087801479224504934912, 5.21561954477899575985853167184, 5.96543580075354110580505048215, 6.48484357198194845975797217823, 7.32826615231018273816399325592, 7.59918729163757270369104716022, 8.376385804842880572823578036385, 8.605693869432553999354344351766, 9.286990966849537833715460133347, 9.797102992710542232518309984184, 10.26906316462456696937320413075

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