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Properties

Label 4-5200e2-1.1-c1e2-0-5
Degree $4$
Conductor $27040000$
Sign $1$
Analytic cond. $1724.09$
Root an. cond. $6.44377$
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 + 4·7-s + 6·11-s − 2·13-s + 2·19-s + 8·21-s + 6·23-s − 2·27-s − 12·29-s − 10·31-s + 12·33-s + 8·37-s − 4·39-s + 10·43-s + 12·47-s − 2·49-s + 4·57-s + 6·59-s + 4·61-s − 8·67-s + 12·69-s − 6·71-s + 8·73-s + 24·77-s − 4·79-s − 81-s − 12·83-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.51·7-s + 1.80·11-s − 0.554·13-s + 0.458·19-s + 1.74·21-s + 1.25·23-s − 0.384·27-s − 2.22·29-s − 1.79·31-s + 2.08·33-s + 1.31·37-s − 0.640·39-s + 1.52·43-s + 1.75·47-s − 2/7·49-s + 0.529·57-s + 0.781·59-s + 0.512·61-s − 0.977·67-s + 1.44·69-s − 0.712·71-s + 0.936·73-s + 2.73·77-s − 0.450·79-s − 1/9·81-s − 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(27040000\)    =    \(2^{8} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1724.09\)
Root analytic conductor: \(6.44377\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 27040000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.537423767\)
\(L(\frac12)\) \(\approx\) \(6.537423767\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.3.ac_e
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$D_{4}$ \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.11.ag_bc
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.17.a_w
19$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_m
23$D_{4}$ \( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_ca
29$D_{4}$ \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.29.m_de
31$D_{4}$ \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.31.k_ci
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.41.a_cs
43$D_{4}$ \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.43.ak_dg
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.47.am_fa
53$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.53.a_ac
59$D_{4}$ \( 1 - 6 T - 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.59.ag_au
61$D_{4}$ \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.61.ae_s
67$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.67.i_bq
71$D_{4}$ \( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.71.g_fs
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.79.e_cc
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.89.m_gk
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.360967602506322930488406972059, −8.149302797345292283789720749841, −7.56529966381496853131630373242, −7.37402774901200435038193669476, −7.19942446387734033625674059369, −6.83934690187191850691686630739, −6.07081110768405774792031242144, −5.89218354231820530106595115945, −5.41777375243067217362024377236, −5.19250508640671902539408678102, −4.53535921777301995898221240981, −4.28057766992217223619736930245, −3.89461984831902207812229647421, −3.56265407230517161522276877037, −2.98374483108912990385152223901, −2.69336812107681752038983976709, −1.94395657180881001617537923733, −1.85407883614589104412159820560, −1.24822697167071408040488332500, −0.64671634344299014659495817708, 0.64671634344299014659495817708, 1.24822697167071408040488332500, 1.85407883614589104412159820560, 1.94395657180881001617537923733, 2.69336812107681752038983976709, 2.98374483108912990385152223901, 3.56265407230517161522276877037, 3.89461984831902207812229647421, 4.28057766992217223619736930245, 4.53535921777301995898221240981, 5.19250508640671902539408678102, 5.41777375243067217362024377236, 5.89218354231820530106595115945, 6.07081110768405774792031242144, 6.83934690187191850691686630739, 7.19942446387734033625674059369, 7.37402774901200435038193669476, 7.56529966381496853131630373242, 8.149302797345292283789720749841, 8.360967602506322930488406972059

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