Properties

Label 4-999e2-1.1-c1e2-0-11
Degree $4$
Conductor $998001$
Sign $1$
Analytic cond. $63.6334$
Root an. cond. $2.82436$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·4-s + 2·7-s + 12·16-s + 10·25-s + 8·28-s + 11·37-s − 11·49-s + 32·64-s − 10·67-s + 14·73-s + 40·100-s + 24·112-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 44·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 173-s + 20·175-s + 179-s + ⋯
L(s)  = 1  + 2·4-s + 0.755·7-s + 3·16-s + 2·25-s + 1.51·28-s + 1.80·37-s − 1.57·49-s + 4·64-s − 1.22·67-s + 1.63·73-s + 4·100-s + 2.26·112-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.61·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.0769·169-s + 0.0760·173-s + 1.51·175-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 998001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 998001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(998001\)    =    \(3^{6} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(63.6334\)
Root analytic conductor: \(2.82436\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{999} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 998001,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.617821610\)
\(L(\frac12)\) \(\approx\) \(4.617821610\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
37$C_2$ \( 1 - 11 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T^{2} )^{2} \)
5$C_2$ \( ( 1 - p T^{2} )^{2} \)
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - p T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
67$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \)
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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

−10.25113730838457757421992632202, −10.01165935286159240033703753940, −9.334563920871697063557719945381, −8.970708775727529334572320771536, −8.191005004300663282348888984927, −8.168466974740101532234017702387, −7.56377311378779453474788841346, −7.32057804120496983062513471907, −6.65713911762105507784367158492, −6.55537936853374091652678039759, −5.99968177963190436725065673454, −5.59969311687957729625074763979, −4.82854933191248840480641419749, −4.74708859495628763244906057756, −3.73130298085626340887296770287, −3.31258171165782545655249386174, −2.55854356271730059640694896368, −2.45181260533058921939057205942, −1.49074209117281793383131546920, −1.10707738219417294326604830125, 1.10707738219417294326604830125, 1.49074209117281793383131546920, 2.45181260533058921939057205942, 2.55854356271730059640694896368, 3.31258171165782545655249386174, 3.73130298085626340887296770287, 4.74708859495628763244906057756, 4.82854933191248840480641419749, 5.59969311687957729625074763979, 5.99968177963190436725065673454, 6.55537936853374091652678039759, 6.65713911762105507784367158492, 7.32057804120496983062513471907, 7.56377311378779453474788841346, 8.168466974740101532234017702387, 8.191005004300663282348888984927, 8.970708775727529334572320771536, 9.334563920871697063557719945381, 10.01165935286159240033703753940, 10.25113730838457757421992632202

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