Properties

Label 4-416000-1.1-c1e2-0-1
Degree $4$
Conductor $416000$
Sign $1$
Analytic cond. $26.5245$
Root an. cond. $2.26940$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5-s + 9-s + 4·13-s − 3·15-s + 25-s + 12·27-s + 11·31-s − 10·37-s − 12·39-s + 5·41-s + 45-s − 3·49-s − 13·53-s + 4·65-s + 15·71-s − 3·75-s + 21·79-s − 26·81-s − 4·83-s − 89-s − 33·93-s − 12·107-s + 30·111-s + 4·117-s + 4·121-s − 15·123-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 1/3·9-s + 1.10·13-s − 0.774·15-s + 1/5·25-s + 2.30·27-s + 1.97·31-s − 1.64·37-s − 1.92·39-s + 0.780·41-s + 0.149·45-s − 3/7·49-s − 1.78·53-s + 0.496·65-s + 1.78·71-s − 0.346·75-s + 2.36·79-s − 2.88·81-s − 0.439·83-s − 0.105·89-s − 3.42·93-s − 1.16·107-s + 2.84·111-s + 0.369·117-s + 4/11·121-s − 1.35·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(416000\)    =    \(2^{8} \cdot 5^{3} \cdot 13\)
Sign: $1$
Analytic conductor: \(26.5245\)
Root analytic conductor: \(2.26940\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 416000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8691340590\)
\(L(\frac12)\) \(\approx\) \(0.8691340590\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 5 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.d_i
7$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \) 2.7.a_d
11$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.11.a_ae
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.31.al_di
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.k_dm
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.41.af_de
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.a_cs
47$C_2^2$ \( 1 + 21 T^{2} + p^{2} T^{4} \) 2.47.a_v
53$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.53.n_fm
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.59.a_u
61$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.61.a_acv
67$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.67.a_ev
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.71.ap_gw
73$C_2^2$ \( 1 - 71 T^{2} + p^{2} T^{4} \) 2.73.a_act
79$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.79.av_je
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.83.e_go
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) 2.89.b_gw
97$C_2^2$ \( 1 - 181 T^{2} + p^{2} T^{4} \) 2.97.a_agz
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

−8.536537348694070494538693926296, −8.216151311021327596394724806325, −7.81981217837061270058475330772, −6.85004279464337364104067735528, −6.61707567366494113881741510716, −6.26682284690255028903527535205, −5.83297054512272774313736487742, −5.43835818889124127021163953680, −4.89718146667876453390051759152, −4.60168721224253558559374544259, −3.69529143253407862403454473374, −3.13604368683873135394589058924, −2.41822004341774326990898921562, −1.41009488449034912040712289684, −0.60640883161006825396017442901, 0.60640883161006825396017442901, 1.41009488449034912040712289684, 2.41822004341774326990898921562, 3.13604368683873135394589058924, 3.69529143253407862403454473374, 4.60168721224253558559374544259, 4.89718146667876453390051759152, 5.43835818889124127021163953680, 5.83297054512272774313736487742, 6.26682284690255028903527535205, 6.61707567366494113881741510716, 6.85004279464337364104067735528, 7.81981217837061270058475330772, 8.216151311021327596394724806325, 8.536537348694070494538693926296

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