↑

Properties

Label 4-1001e2-1.1-c1e2-0-2
Degree $4$
Conductor $1002001$
Sign $-1$
Analytic cond. $63.8884$
Root an. cond. $2.82719$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·3-s − 4·4-s − 6·5-s + 6·9-s + 16·12-s + 24·15-s + 12·16-s + 24·20-s + 6·23-s + 17·25-s + 4·27-s + 10·31-s − 24·36-s + 4·37-s − 36·45-s + 6·47-s − 48·48-s + 49-s − 18·53-s − 96·60-s − 32·64-s + 28·67-s − 24·69-s − 12·71-s − 68·75-s − 72·80-s − 37·81-s + ⋯
L(s)  = 1  − 2.30·3-s − 2·4-s − 2.68·5-s + 2·9-s + 4.61·12-s + 6.19·15-s + 3·16-s + 5.36·20-s + 1.25·23-s + 17/5·25-s + 0.769·27-s + 1.79·31-s − 4·36-s + 0.657·37-s − 5.36·45-s + 0.875·47-s − 6.92·48-s + 1/7·49-s − 2.47·53-s − 12.3·60-s − 4·64-s + 3.42·67-s − 2.88·69-s − 1.42·71-s − 7.85·75-s − 8.04·80-s − 4.11·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1002001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1002001\)    =    \(7^{2} \cdot 11^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(63.8884\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1002001,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.2.a_e
3$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.3.e_k
5$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.5.g_t
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.19.a_al
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.29.a_ax
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.43.a_dh
47$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.47.ag_dz
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.67.abc_ms
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.73.a_z
79$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.79.a_gb
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.83.a_gb
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \) 2.89.abe_pn
97$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.97.c_hn
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.023034081311283559976527256533, −7.60332298682950357021318080507, −6.97556725662558611504386870793, −6.48432181419363077598714474936, −6.05510665689561164799845565623, −5.47928828118502604201716906159, −4.95454046107923865998450494116, −4.82094210358198316082444203410, −4.36526056113723032255916840174, −4.04609643315305459582542236808, −3.36854503826664578071054317862, −3.00675363687601775920051816088, −0.883826827840201941525555392849, −0.71239963936312711175952583959, 0, 0.71239963936312711175952583959, 0.883826827840201941525555392849, 3.00675363687601775920051816088, 3.36854503826664578071054317862, 4.04609643315305459582542236808, 4.36526056113723032255916840174, 4.82094210358198316082444203410, 4.95454046107923865998450494116, 5.47928828118502604201716906159, 6.05510665689561164799845565623, 6.48432181419363077598714474936, 6.97556725662558611504386870793, 7.60332298682950357021318080507, 8.023034081311283559976527256533

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