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Properties

Label 4-2240e2-1.1-c1e2-0-3
Degree $4$
Conductor $5017600$
Sign $1$
Analytic cond. $319.926$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·5-s − 2·7-s + 3·9-s − 7·11-s − 3·13-s + 2·15-s + 5·17-s − 2·19-s − 2·21-s + 2·23-s + 3·25-s + 8·27-s + 3·29-s − 16·31-s − 7·33-s − 4·35-s + 4·37-s − 3·39-s + 2·41-s + 6·43-s + 6·45-s + 3·47-s + 3·49-s + 5·51-s − 10·53-s − 14·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.894·5-s − 0.755·7-s + 9-s − 2.11·11-s − 0.832·13-s + 0.516·15-s + 1.21·17-s − 0.458·19-s − 0.436·21-s + 0.417·23-s + 3/5·25-s + 1.53·27-s + 0.557·29-s − 2.87·31-s − 1.21·33-s − 0.676·35-s + 0.657·37-s − 0.480·39-s + 0.312·41-s + 0.914·43-s + 0.894·45-s + 0.437·47-s + 3/7·49-s + 0.700·51-s − 1.37·53-s − 1.88·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5017600\)    =    \(2^{12} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(319.926\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5017600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.327920017\)
\(L(\frac12)\) \(\approx\) \(2.327920017\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$D_{4}$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.11.h_ba
13$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_u
17$D_{4}$ \( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bg
19$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.19.c_g
23$D_{4}$ \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.23.ac_o
29$D_{4}$ \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.29.ad_ca
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$D_{4}$ \( 1 - 2 T + 50 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.41.ac_by
43$D_{4}$ \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ck
47$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_w
53$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_du
59$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.59.q_ha
61$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_du
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$D_{4}$ \( 1 - 13 T + 126 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.79.an_ew
83$D_{4}$ \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_bm
89$D_{4}$ \( 1 - 18 T + 226 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_is
97$D_{4}$ \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.97.aj_i
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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

−9.395974814950879083279226335886, −8.965188769327273688629591428896, −8.455236411332211205886319644615, −7.962701631456653947542615130948, −7.57490923843974170141836969711, −7.49301554514809412248307777584, −7.01545349556250671686235454755, −6.53010976196831502227079732817, −6.00917397781434987084747583413, −5.72820970159234419898133277402, −5.23589701963254618285651913701, −4.84469293976948923208492079587, −4.59467857619858477638165897048, −3.82069093593627035379042340785, −3.18688025905670378389089845461, −3.08949764723252265928376375935, −2.33570628636096095604892261099, −2.17898061949279373872554101835, −1.38696184868339864919422400914, −0.49933168956732824256766810596, 0.49933168956732824256766810596, 1.38696184868339864919422400914, 2.17898061949279373872554101835, 2.33570628636096095604892261099, 3.08949764723252265928376375935, 3.18688025905670378389089845461, 3.82069093593627035379042340785, 4.59467857619858477638165897048, 4.84469293976948923208492079587, 5.23589701963254618285651913701, 5.72820970159234419898133277402, 6.00917397781434987084747583413, 6.53010976196831502227079732817, 7.01545349556250671686235454755, 7.49301554514809412248307777584, 7.57490923843974170141836969711, 7.962701631456653947542615130948, 8.455236411332211205886319644615, 8.965188769327273688629591428896, 9.395974814950879083279226335886

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