↑

Properties

Label 4-9300e2-1.1-c1e2-0-10
Degree $4$
Conductor $86490000$
Sign $1$
Analytic cond. $5514.67$
Root an. cond. $8.61747$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 3·9-s + 11-s + 4·17-s − 5·19-s − 5·23-s − 4·27-s − 2·29-s − 2·31-s − 2·33-s + 2·37-s − 6·41-s + 13·43-s + 21·47-s − 14·49-s − 8·51-s + 53-s + 10·57-s − 8·59-s − 12·61-s − 7·67-s + 10·69-s + 3·71-s − 12·73-s + 7·79-s + 5·81-s − 3·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 9-s + 0.301·11-s + 0.970·17-s − 1.14·19-s − 1.04·23-s − 0.769·27-s − 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.328·37-s − 0.937·41-s + 1.98·43-s + 3.06·47-s − 2·49-s − 1.12·51-s + 0.137·53-s + 1.32·57-s − 1.04·59-s − 1.53·61-s − 0.855·67-s + 1.20·69-s + 0.356·71-s − 1.40·73-s + 0.787·79-s + 5/9·81-s − 0.329·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86490000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(5514.67\)
Root analytic conductor: \(8.61747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 86490000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.7.a_o
11$D_{4}$ \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) 2.11.ab_o
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$D_{4}$ \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_f
19$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.19.f_bk
23$D_{4}$ \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bs
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.29.c_ch
37$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_bq
41$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_cg
43$D_{4}$ \( 1 - 13 T + 120 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.43.an_eq
47$D_{4}$ \( 1 - 21 T + 196 T^{2} - 21 p T^{3} + p^{2} T^{4} \) 2.47.av_ho
53$D_{4}$ \( 1 - T + 32 T^{2} - p T^{3} + p^{2} T^{4} \) 2.53.ab_bg
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.61.m_gc
67$D_{4}$ \( 1 + 7 T + 138 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.67.h_fi
71$D_{4}$ \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.71.ad_fg
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$D_{4}$ \( 1 - 7 T + 96 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.79.ah_ds
83$D_{4}$ \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.83.d_abm
89$D_{4}$ \( 1 + 24 T + 289 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.89.y_ld
97$D_{4}$ \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.97.ac_cl
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

−7.40016046654567446455739482227, −7.35940670143531486036033503556, −6.66848234665000314611266760640, −6.53011170020070476804755653983, −6.06694682630721436939378343220, −5.88694941618317879181133425727, −5.46367183542135324355916901275, −5.40636808905250834905626488989, −4.58358824758331500756759069811, −4.50336916867751793843462825548, −4.09683759400671147974491126491, −3.85458851992385564249353471884, −3.22625897250492789781048583843, −2.90744284943922985168594411279, −2.24791033977786899452112330722, −1.95805926900726843985347851028, −1.24994265043177271723390979968, −1.09385669786786343451101122301, 0, 0, 1.09385669786786343451101122301, 1.24994265043177271723390979968, 1.95805926900726843985347851028, 2.24791033977786899452112330722, 2.90744284943922985168594411279, 3.22625897250492789781048583843, 3.85458851992385564249353471884, 4.09683759400671147974491126491, 4.50336916867751793843462825548, 4.58358824758331500756759069811, 5.40636808905250834905626488989, 5.46367183542135324355916901275, 5.88694941618317879181133425727, 6.06694682630721436939378343220, 6.53011170020070476804755653983, 6.66848234665000314611266760640, 7.35940670143531486036033503556, 7.40016046654567446455739482227

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