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Properties

Label 4-1400e2-1.1-c1e2-0-33
Degree $4$
Conductor $1960000$
Sign $1$
Analytic cond. $124.971$
Root an. cond. $3.34350$
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 + 5·7-s + 3·9-s − 2·11-s + 4·17-s + 2·19-s + 5·21-s + 23-s + 8·27-s + 18·29-s − 4·31-s − 2·33-s + 4·37-s + 2·41-s − 18·43-s + 18·49-s + 4·51-s − 10·53-s + 2·57-s + 10·59-s − 9·61-s + 15·63-s + 5·67-s + 69-s + 28·71-s + 12·73-s − 10·77-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.88·7-s + 9-s − 0.603·11-s + 0.970·17-s + 0.458·19-s + 1.09·21-s + 0.208·23-s + 1.53·27-s + 3.34·29-s − 0.718·31-s − 0.348·33-s + 0.657·37-s + 0.312·41-s − 2.74·43-s + 18/7·49-s + 0.560·51-s − 1.37·53-s + 0.264·57-s + 1.30·59-s − 1.15·61-s + 1.88·63-s + 0.610·67-s + 0.120·69-s + 3.32·71-s + 1.40·73-s − 1.13·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1960000\)    =    \(2^{6} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(124.971\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1960000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.767392848\)
\(L(\frac12)\) \(\approx\) \(4.767392848\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
good3$C_2^2$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_ac
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_ab
19$C_2^2$ \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ap
23$C_2^2$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) 2.23.ab_aw
29$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.29.as_fj
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.31.e_ap
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.37.ae_av
41$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.41.ac_df
43$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.43.s_gl
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2^2$ \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.53.k_bv
59$C_2^2$ \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.59.ak_bp
61$C_2^2$ \( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.61.j_u
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.67.af_abq
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.71.abc_na
73$C_2^2$ \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.73.am_ct
79$C_2^2$ \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.79.o_en
83$C_2$ \( ( 1 + 11 T + p T^{2} )^{2} \) 2.83.w_lb
89$C_2^2$ \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) 2.89.ap_fg
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_ty
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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

−9.652592586863270284937053401834, −9.602096546989072272313445434362, −8.590147594128129468219724229426, −8.454955017668562899855881730591, −8.316662137544405238704271389936, −7.88011769431113586635856670531, −7.29606090114691202928113893854, −7.22532539764096931725229853673, −6.44146316349184135358619177274, −6.24806869283686436469761587881, −5.33528887498942377122718255139, −5.01188586371534652107990766585, −4.76538982303696706437450423522, −4.47710756627370385212212925778, −3.58783644984351974046766465919, −3.31257159205927625510424581959, −2.52196403544191999001207518424, −2.17357505693376434966615532216, −1.19560545046866331209601796830, −1.11703388513479410951221812456, 1.11703388513479410951221812456, 1.19560545046866331209601796830, 2.17357505693376434966615532216, 2.52196403544191999001207518424, 3.31257159205927625510424581959, 3.58783644984351974046766465919, 4.47710756627370385212212925778, 4.76538982303696706437450423522, 5.01188586371534652107990766585, 5.33528887498942377122718255139, 6.24806869283686436469761587881, 6.44146316349184135358619177274, 7.22532539764096931725229853673, 7.29606090114691202928113893854, 7.88011769431113586635856670531, 8.316662137544405238704271389936, 8.454955017668562899855881730591, 8.590147594128129468219724229426, 9.602096546989072272313445434362, 9.652592586863270284937053401834

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