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Properties

Label 4-1200e2-1.1-c1e2-0-5
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $91.8156$
Root an. cond. $3.09548$
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  − 9-s − 4·11-s − 14·19-s + 4·29-s + 10·31-s + 24·41-s + 5·49-s − 12·59-s − 26·61-s + 8·71-s − 16·79-s + 81-s − 32·89-s + 4·99-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 14·171-s + ⋯
L(s)  = 1  − 1/3·9-s − 1.20·11-s − 3.21·19-s + 0.742·29-s + 1.79·31-s + 3.74·41-s + 5/7·49-s − 1.56·59-s − 3.32·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 3.39·89-s + 0.402·99-s − 1.72·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + 1.07·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(91.8156\)
Root analytic conductor: \(3.09548\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.032242198\)
\(L(\frac12)\) \(\approx\) \(1.032242198\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.13.a_ar
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.19.o_dj
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.31.ak_dj
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.41.ay_is
43$C_2^2$ \( 1 - 77 T^{2} + p^{2} T^{4} \) 2.43.a_acz
47$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.47.a_g
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.59.m_fy
61$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.61.ba_lf
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.67.a_adh
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.89.bg_qs
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \) 2.97.a_afp
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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

−10.11644947816288021049818698252, −9.314537063924962042277544031807, −9.288766405770784443349619384584, −8.615875208796458033587449771537, −8.319619477119805774047537149854, −7.934135284358977194002669909799, −7.70168657859402142766385711202, −7.00109256722664370375633776604, −6.58686196217540139420012569517, −6.07881883372863602955411312406, −5.92096829976934129275414026984, −5.39388365771900559385414862261, −4.47224299455473156134775076265, −4.39262977784870508815624101530, −4.21816388347545143103778305578, −2.97777678868117345492245115772, −2.76349451288150143847834960080, −2.33834021004731874521058085078, −1.51692702799655189650057638228, −0.42800494081046953701531497256, 0.42800494081046953701531497256, 1.51692702799655189650057638228, 2.33834021004731874521058085078, 2.76349451288150143847834960080, 2.97777678868117345492245115772, 4.21816388347545143103778305578, 4.39262977784870508815624101530, 4.47224299455473156134775076265, 5.39388365771900559385414862261, 5.92096829976934129275414026984, 6.07881883372863602955411312406, 6.58686196217540139420012569517, 7.00109256722664370375633776604, 7.70168657859402142766385711202, 7.934135284358977194002669909799, 8.319619477119805774047537149854, 8.615875208796458033587449771537, 9.288766405770784443349619384584, 9.314537063924962042277544031807, 10.11644947816288021049818698252

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