Properties

Label 4-1331200-1.1-c1e2-0-5
Degree $4$
Conductor $1331200$
Sign $1$
Analytic cond. $84.8784$
Root an. cond. $3.03528$
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  − 2·9-s − 3·13-s + 25-s + 20·37-s − 12·41-s + 2·49-s − 12·53-s + 8·61-s + 4·73-s − 5·81-s + 4·97-s − 12·101-s − 4·109-s + 12·113-s + 6·117-s − 10·121-s + ⋯
L(s)  = 1  − 2/3·9-s − 0.832·13-s + 1/5·25-s + 3.28·37-s − 1.87·41-s + 2/7·49-s − 1.64·53-s + 1.02·61-s + 0.468·73-s − 5/9·81-s + 0.406·97-s − 1.19·101-s − 0.383·109-s + 1.12·113-s + 0.554·117-s − 0.909·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1331200\)    =    \(2^{12} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(84.8784\)
Root analytic conductor: \(3.03528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1331200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.446424644\)
\(L(\frac12)\) \(\approx\) \(1.446424644\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 2 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.11.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.19.a_ao
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.43.a_bi
47$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.a_aby
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.59.a_ack
61$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ai_dy
67$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.67.a_w
71$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.71.a_aba
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.73.ae_fu
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.79.a_aby
83$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.83.a_aby
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.ae_cc
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

−8.120941466024761648376868264193, −7.51894186370373817288894540294, −7.17869728007603393644443256733, −6.59414670862135180742642456239, −6.27598849294237011183332610096, −5.77891938082582212698215408254, −5.33668902993406213404835813366, −4.80638857920028277766610591771, −4.47240378955962257838335190328, −3.86694776879159911578148977120, −3.23078955668475626403898725621, −2.74395621140599441023853419876, −2.29901503488163657198627151959, −1.48708270926286920841817472044, −0.52966297078085901064040903571, 0.52966297078085901064040903571, 1.48708270926286920841817472044, 2.29901503488163657198627151959, 2.74395621140599441023853419876, 3.23078955668475626403898725621, 3.86694776879159911578148977120, 4.47240378955962257838335190328, 4.80638857920028277766610591771, 5.33668902993406213404835813366, 5.77891938082582212698215408254, 6.27598849294237011183332610096, 6.59414670862135180742642456239, 7.17869728007603393644443256733, 7.51894186370373817288894540294, 8.120941466024761648376868264193

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