↑

Properties

Label 4-7488e2-1.1-c1e2-0-5
Degree $4$
Conductor $56070144$
Sign $1$
Analytic cond. $3575.08$
Root an. cond. $7.73252$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·7-s + 4·11-s − 2·13-s + 8·17-s − 4·19-s − 8·23-s − 2·25-s − 8·29-s + 4·31-s − 4·37-s + 8·41-s + 4·47-s + 6·49-s − 16·53-s − 4·59-s + 4·61-s + 4·67-s − 12·71-s + 12·73-s + 16·77-s + 8·79-s − 12·83-s + 8·89-s − 8·91-s + 12·97-s + 24·103-s + 8·107-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20·11-s − 0.554·13-s + 1.94·17-s − 0.917·19-s − 1.66·23-s − 2/5·25-s − 1.48·29-s + 0.718·31-s − 0.657·37-s + 1.24·41-s + 0.583·47-s + 6/7·49-s − 2.19·53-s − 0.520·59-s + 0.512·61-s + 0.488·67-s − 1.42·71-s + 1.40·73-s + 1.82·77-s + 0.900·79-s − 1.31·83-s + 0.847·89-s − 0.838·91-s + 1.21·97-s + 2.36·103-s + 0.773·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56070144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(56070144\)    =    \(2^{12} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(3575.08\)
Root analytic conductor: \(7.73252\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 56070144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.633246679\)
\(L(\frac12)\) \(\approx\) \(3.633246679\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_4$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.7.ae_k
11$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_s
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$D_{4}$ \( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.19.e_bi
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.23.i_ck
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$C_4$ \( 1 - 4 T - 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.31.ae_ag
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$D_{4}$ \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.41.ai_dm
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.43.a_cc
47$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_ba
53$D_{4}$ \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.53.q_fi
59$D_{4}$ \( 1 + 4 T + 114 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_ek
61$D_{4}$ \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_ac
67$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.67.ae_fa
71$D_{4}$ \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_ec
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.79.ai_gs
83$D_{4}$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hm
89$D_{4}$ \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.89.ai_es
97$D_{4}$ \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.97.am_dy
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

−7.88368632550861341118424097899, −7.87257744094845082926064793025, −7.39868155815422606742715521929, −7.27726763694587265967943284083, −6.53515788490221402349036629522, −6.34709445877041451921360901962, −5.85945012390542077235290694499, −5.75239784173965392161427851198, −5.21108954494631803039115143318, −4.91812019919999263046098770431, −4.33282036150045662228617801630, −4.32131499598604637052299611472, −3.75150011422070180345507027864, −3.44149246765790979365495193581, −2.99516214667113635219311628017, −2.26349649342322279889454073517, −1.85357920105956979689123341914, −1.73183976765087867153960350613, −1.08664055703967111880159582385, −0.48267628399271911089286539513, 0.48267628399271911089286539513, 1.08664055703967111880159582385, 1.73183976765087867153960350613, 1.85357920105956979689123341914, 2.26349649342322279889454073517, 2.99516214667113635219311628017, 3.44149246765790979365495193581, 3.75150011422070180345507027864, 4.32131499598604637052299611472, 4.33282036150045662228617801630, 4.91812019919999263046098770431, 5.21108954494631803039115143318, 5.75239784173965392161427851198, 5.85945012390542077235290694499, 6.34709445877041451921360901962, 6.53515788490221402349036629522, 7.27726763694587265967943284083, 7.39868155815422606742715521929, 7.87257744094845082926064793025, 7.88368632550861341118424097899

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