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Properties

Label 4-800e2-1.1-c1e2-0-39
Degree $4$
Conductor $640000$
Sign $-1$
Analytic cond. $40.8069$
Root an. cond. $2.52745$
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  + 9-s − 10·41-s − 6·49-s − 20·61-s − 8·81-s − 10·89-s − 4·101-s − 12·109-s − 17·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯
L(s)  = 1  + 1/3·9-s − 1.56·41-s − 6/7·49-s − 2.56·61-s − 8/9·81-s − 1.05·89-s − 0.398·101-s − 1.14·109-s − 1.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(640000\)    =    \(2^{10} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(40.8069\)
Root analytic conductor: \(2.52745\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 640000,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
good3$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.3.a_ab
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \) 2.11.a_r
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.17.a_ap
19$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \) 2.19.a_ah
23$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.23.a_aba
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.31.a_bq
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.41.k_ed
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.59.a_bm
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 129 T^{2} + p^{2} T^{4} \) 2.67.a_aez
71$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.71.a_ck
73$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.73.a_cn
79$C_2^2$ \( 1 + 138 T^{2} + p^{2} T^{4} \) 2.79.a_fi
83$C_2^2$ \( 1 - 41 T^{2} + p^{2} T^{4} \) 2.83.a_abp
89$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.89.k_hv
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.191885967005418230315017742284, −7.68695100998181927378561653905, −7.30436043712554767760269783525, −6.63985268498157670489757081195, −6.55995185094210099396481319097, −5.78960882978330692016400047625, −5.44407312908855453737353449411, −4.79797755876298238088390827546, −4.44310525944967884411517156031, −3.84034847354246775317606382075, −3.21534498987022356609159191884, −2.76518248470981251211555712333, −1.86474429155852563589240025824, −1.32263590099227303284047774219, 0, 1.32263590099227303284047774219, 1.86474429155852563589240025824, 2.76518248470981251211555712333, 3.21534498987022356609159191884, 3.84034847354246775317606382075, 4.44310525944967884411517156031, 4.79797755876298238088390827546, 5.44407312908855453737353449411, 5.78960882978330692016400047625, 6.55995185094210099396481319097, 6.63985268498157670489757081195, 7.30436043712554767760269783525, 7.68695100998181927378561653905, 8.191885967005418230315017742284

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