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Properties

Label 4-3240e2-1.1-c1e2-0-19
Degree $4$
Conductor $10497600$
Sign $1$
Analytic cond. $669.336$
Root an. cond. $5.08640$
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  − 5-s − 2·11-s + 6·17-s − 2·19-s − 3·23-s − 4·29-s + 5·31-s + 20·37-s − 6·41-s + 6·43-s − 8·47-s + 7·49-s + 6·53-s + 2·55-s − 5·61-s + 2·67-s + 4·71-s + 12·73-s + 11·79-s − 9·83-s − 6·85-s + 20·89-s + 2·95-s − 8·97-s + 12·101-s + 12·103-s − 8·107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.603·11-s + 1.45·17-s − 0.458·19-s − 0.625·23-s − 0.742·29-s + 0.898·31-s + 3.28·37-s − 0.937·41-s + 0.914·43-s − 1.16·47-s + 49-s + 0.824·53-s + 0.269·55-s − 0.640·61-s + 0.244·67-s + 0.474·71-s + 1.40·73-s + 1.23·79-s − 0.987·83-s − 0.650·85-s + 2.11·89-s + 0.205·95-s − 0.812·97-s + 1.19·101-s + 1.18·103-s − 0.773·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10497600\)    =    \(2^{6} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(669.336\)
Root analytic conductor: \(5.08640\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10497600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.610463609\)
\(L(\frac12)\) \(\approx\) \(2.610463609\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 + T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.13.a_an
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.17.ag_br
19$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.19.c_bn
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.23.d_ao
29$C_2^2$ \( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.29.e_an
31$C_2^2$ \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.31.af_ag
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.37.au_gs
41$C_2^2$ \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_af
43$C_2^2$ \( 1 - 6 T - 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.43.ag_ah
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_r
53$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.53.ag_el
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.61.f_abk
67$C_2^2$ \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.67.ac_acl
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.79.al_bq
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.83.j_ac
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.97.i_abh
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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

−8.699157407487801783117826202595, −8.375644653711369405055821968656, −7.949927499266847783744645699148, −7.895713173417943716693383897678, −7.35312361971692527748573688796, −7.21259472537554617289025941703, −6.43557297735975774945411042782, −6.25002722702370821511111696866, −5.73163655044939700886347902003, −5.61619642531035342431065107657, −4.82008112447506116388268978524, −4.73146679433286267611047752682, −4.10288208409951118958109915722, −3.82710629090034865094885928173, −3.10984295732141777961756380764, −3.04917901351017251881082674340, −2.09177287399667536600513909418, −2.07715463233027668175503528762, −0.885367002401265269664088565825, −0.65920071246210068966079514178, 0.65920071246210068966079514178, 0.885367002401265269664088565825, 2.07715463233027668175503528762, 2.09177287399667536600513909418, 3.04917901351017251881082674340, 3.10984295732141777961756380764, 3.82710629090034865094885928173, 4.10288208409951118958109915722, 4.73146679433286267611047752682, 4.82008112447506116388268978524, 5.61619642531035342431065107657, 5.73163655044939700886347902003, 6.25002722702370821511111696866, 6.43557297735975774945411042782, 7.21259472537554617289025941703, 7.35312361971692527748573688796, 7.895713173417943716693383897678, 7.949927499266847783744645699148, 8.375644653711369405055821968656, 8.699157407487801783117826202595

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