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Properties

Label 4-7440e2-1.1-c1e2-0-2
Degree $4$
Conductor $55353600$
Sign $1$
Analytic cond. $3529.39$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 7-s + 3·9-s − 5·11-s + 12·13-s + 4·15-s − 8·17-s − 19-s − 2·21-s − 5·23-s + 3·25-s − 4·27-s − 2·31-s + 10·33-s − 2·35-s + 2·37-s − 24·39-s − 2·41-s + 9·43-s − 6·45-s − 6·47-s + 3·49-s + 16·51-s + 3·53-s + 10·55-s + 2·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.377·7-s + 9-s − 1.50·11-s + 3.32·13-s + 1.03·15-s − 1.94·17-s − 0.229·19-s − 0.436·21-s − 1.04·23-s + 3/5·25-s − 0.769·27-s − 0.359·31-s + 1.74·33-s − 0.338·35-s + 0.328·37-s − 3.84·39-s − 0.312·41-s + 1.37·43-s − 0.894·45-s − 0.875·47-s + 3/7·49-s + 2.24·51-s + 0.412·53-s + 1.34·55-s + 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55353600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(55353600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(3529.39\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 55353600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.558859184\)
\(L(\frac12)\) \(\approx\) \(1.558859184\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
31$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.7.ab_ac
11$D_{4}$ \( 1 + 5 T + 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.11.f_m
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$D_{4}$ \( 1 + T + 22 T^{2} + p T^{3} + p^{2} T^{4} \) 2.19.b_w
23$D_{4}$ \( 1 + 5 T + 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.23.f_bk
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
37$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.37.ac_k
41$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_s
43$D_{4}$ \( 1 - 9 T + 90 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.43.aj_dm
47$D_{4}$ \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.47.g_bm
53$D_{4}$ \( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.53.ad_do
59$D_{4}$ \( 1 - 2 T + 54 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.59.ac_cc
61$D_{4}$ \( 1 + 6 T + 66 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_co
67$D_{4}$ \( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_da
71$D_{4}$ \( 1 - 17 T + 198 T^{2} - 17 p T^{3} + p^{2} T^{4} \) 2.71.ar_hq
73$D_{4}$ \( 1 + 9 T + 150 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_fu
79$D_{4}$ \( 1 - 9 T + 162 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.79.aj_gg
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.83.aq_iw
89$D_{4}$ \( 1 + 5 T + 168 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.89.f_gm
97$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.97.a_aco
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.927605281546615105367350370677, −7.88413628994546034935872020793, −7.40866968504177466775200389236, −6.92855033623589933810858705675, −6.48248279831182523318310026697, −6.38910162881556143377113636964, −6.03509687572585569879298899759, −5.68805664370605676058455095284, −5.12451889284970536144608392943, −5.12225486762666260595325324243, −4.34524991014029457215251161546, −4.21340824473023019066644361725, −3.86724623951996945537686081937, −3.57003790848865598584176936002, −2.95713556374044904012814970059, −2.44648792100160808949598212666, −1.83309021459046558231924344309, −1.57423138910224417686506440346, −0.60552204115459185316637828386, −0.56758951038962470531588040708, 0.56758951038962470531588040708, 0.60552204115459185316637828386, 1.57423138910224417686506440346, 1.83309021459046558231924344309, 2.44648792100160808949598212666, 2.95713556374044904012814970059, 3.57003790848865598584176936002, 3.86724623951996945537686081937, 4.21340824473023019066644361725, 4.34524991014029457215251161546, 5.12225486762666260595325324243, 5.12451889284970536144608392943, 5.68805664370605676058455095284, 6.03509687572585569879298899759, 6.38910162881556143377113636964, 6.48248279831182523318310026697, 6.92855033623589933810858705675, 7.40866968504177466775200389236, 7.88413628994546034935872020793, 7.927605281546615105367350370677

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