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Properties

Label 4-80e2-1.1-c1e2-0-6
Degree $4$
Conductor $6400$
Sign $-1$
Analytic cond. $0.408069$
Root an. cond. $0.799251$
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  − 4·3-s + 6·9-s − 12·17-s − 8·19-s + 25-s + 4·27-s + 12·41-s − 20·43-s − 10·49-s + 48·51-s + 32·57-s + 24·59-s + 4·67-s + 4·73-s − 4·75-s − 37·81-s + 12·83-s − 12·89-s + 4·97-s − 12·107-s − 12·113-s − 22·121-s − 48·123-s + ⋯
L(s)  = 1  − 2.30·3-s + 2·9-s − 2.91·17-s − 1.83·19-s + 1/5·25-s + 0.769·27-s + 1.87·41-s − 3.04·43-s − 1.42·49-s + 6.72·51-s + 4.23·57-s + 3.12·59-s + 0.488·67-s + 0.468·73-s − 0.461·75-s − 4.11·81-s + 1.31·83-s − 1.27·89-s + 0.406·97-s − 1.16·107-s − 1.12·113-s − 2·121-s − 4.32·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6400\)    =    \(2^{8} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(0.408069\)
Root analytic conductor: \(0.799251\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 6400,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
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.i_cc
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.43.u_he
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.47.a_cg
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.a_eo
67$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.67.ae_fi
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.97.ae_hq
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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

−11.50854453196989171707939391081, −11.16454648418111686213251777526, −10.83473950301065015207539180976, −10.35239107507603235483123434521, −9.511328653906225471558818278712, −8.552217550781204179646223792184, −8.404829898294941223649273971532, −6.87772159023272676690262517558, −6.57891116465648258947670054106, −6.26903409609752801893873868553, −5.32703134203753576211066117067, −4.78130792717525308450176413839, −4.13045861856120535222472022668, −2.32736297473362010207341859714, 0, 2.32736297473362010207341859714, 4.13045861856120535222472022668, 4.78130792717525308450176413839, 5.32703134203753576211066117067, 6.26903409609752801893873868553, 6.57891116465648258947670054106, 6.87772159023272676690262517558, 8.404829898294941223649273971532, 8.552217550781204179646223792184, 9.511328653906225471558818278712, 10.35239107507603235483123434521, 10.83473950301065015207539180976, 11.16454648418111686213251777526, 11.50854453196989171707939391081

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