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Properties

Label 4-3840e2-1.1-c1e2-0-42
Degree $4$
Conductor $14745600$
Sign $1$
Analytic cond. $940.192$
Root an. cond. $5.53737$
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 + 3·9-s + 12·13-s + 4·15-s − 25-s − 4·27-s + 16·31-s + 20·37-s − 24·39-s + 12·41-s + 8·43-s − 6·45-s − 2·49-s + 12·53-s − 24·65-s + 8·67-s − 16·71-s + 2·75-s − 24·79-s + 5·81-s − 16·83-s + 12·89-s − 32·93-s − 40·111-s + 36·117-s + 22·121-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 9-s + 3.32·13-s + 1.03·15-s − 1/5·25-s − 0.769·27-s + 2.87·31-s + 3.28·37-s − 3.84·39-s + 1.87·41-s + 1.21·43-s − 0.894·45-s − 2/7·49-s + 1.64·53-s − 2.97·65-s + 0.977·67-s − 1.89·71-s + 0.230·75-s − 2.70·79-s + 5/9·81-s − 1.75·83-s + 1.27·89-s − 3.31·93-s − 3.79·111-s + 3.32·117-s + 2·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14745600\)    =    \(2^{16} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(940.192\)
Root analytic conductor: \(5.53737\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14745600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.379614999\)
\(L(\frac12)\) \(\approx\) \(2.379614999\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.7.a_c
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.29.a_g
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
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.am_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \) 2.47.a_adm
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.59.a_ba
61$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.61.a_cw
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.q_hy
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.79.y_lq
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.83.q_iw
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−8.411772705319067576721238517919, −8.379916880006925162472316143505, −7.967314124798459046028881949536, −7.61968180557783570759782479667, −7.16181638274302963968502988478, −6.78110007517117028618950061742, −6.17265951842704904426305040311, −6.05094770933063583748231132730, −5.91296591980457636011296854570, −5.59906536980776722814129760033, −4.69836325677613360223673500981, −4.36481820466417153689628669151, −4.12793775552164044992206194290, −3.99136953022986276516737174272, −3.23268003449758948079940479413, −2.84615067839904685560346464794, −2.28126983532746710351164822964, −1.24434322889413472729196767475, −1.04304104692219698330732547540, −0.66948686774288900859461338694, 0.66948686774288900859461338694, 1.04304104692219698330732547540, 1.24434322889413472729196767475, 2.28126983532746710351164822964, 2.84615067839904685560346464794, 3.23268003449758948079940479413, 3.99136953022986276516737174272, 4.12793775552164044992206194290, 4.36481820466417153689628669151, 4.69836325677613360223673500981, 5.59906536980776722814129760033, 5.91296591980457636011296854570, 6.05094770933063583748231132730, 6.17265951842704904426305040311, 6.78110007517117028618950061742, 7.16181638274302963968502988478, 7.61968180557783570759782479667, 7.967314124798459046028881949536, 8.379916880006925162472316143505, 8.411772705319067576721238517919

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