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Properties

Label 4-72e4-1.1-c1e2-0-5
Degree $4$
Conductor $26873856$
Sign $1$
Analytic cond. $1713.50$
Root an. cond. $6.43385$
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  + 2·5-s − 6·13-s − 10·17-s − 7·25-s + 10·29-s + 10·37-s + 4·41-s + 10·49-s + 4·53-s + 26·61-s − 12·65-s + 6·73-s − 20·85-s − 26·89-s − 12·97-s + 20·101-s + 18·109-s − 2·113-s + 2·121-s − 26·125-s + 127-s + 131-s + 137-s + 139-s + 20·145-s + 149-s + 151-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.66·13-s − 2.42·17-s − 7/5·25-s + 1.85·29-s + 1.64·37-s + 0.624·41-s + 10/7·49-s + 0.549·53-s + 3.32·61-s − 1.48·65-s + 0.702·73-s − 2.16·85-s − 2.75·89-s − 1.21·97-s + 1.99·101-s + 1.72·109-s − 0.188·113-s + 2/11·121-s − 2.32·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.66·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(26873856\)    =    \(2^{12} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(1713.50\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 26873856,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.976799280\)
\(L(\frac12)\) \(\approx\) \(1.976799280\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.5.ac_l
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.13.g_bj
17$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.17.k_ch
19$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.19.a_o
23$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.23.a_w
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.29.ak_df
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.43.a_ck
47$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.47.a_ac
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.59.a_w
61$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \) 2.61.aba_lf
67$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.67.a_eg
71$C_2^2$ \( 1 + 118 T^{2} + p^{2} T^{4} \) 2.71.a_eo
73$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.73.ag_fz
79$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.79.a_acg
83$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.83.a_cs
89$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.89.ba_nj
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.97.m_iw
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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

−8.292561034955820572303942187335, −8.200386287956119303385165045666, −7.52792909449380972553292741129, −7.35016128077794952947756445393, −6.86560675703035315582603165055, −6.64814487452146496416253723123, −6.23738399037541235003412178087, −5.95225691506723163062900247254, −5.32108269438731982573621216632, −5.31237207793323317799404399960, −4.61127650478185954115680499246, −4.37321541019385581044495728266, −4.10128762956455723781952622178, −3.59538203879909240681181916106, −2.64006260900962957738422167489, −2.56321486375426468301571362903, −2.30273994905481870817493677741, −1.87039303864702649218107977830, −1.01549214196947331144143552609, −0.40236422966499168881261365711, 0.40236422966499168881261365711, 1.01549214196947331144143552609, 1.87039303864702649218107977830, 2.30273994905481870817493677741, 2.56321486375426468301571362903, 2.64006260900962957738422167489, 3.59538203879909240681181916106, 4.10128762956455723781952622178, 4.37321541019385581044495728266, 4.61127650478185954115680499246, 5.31237207793323317799404399960, 5.32108269438731982573621216632, 5.95225691506723163062900247254, 6.23738399037541235003412178087, 6.64814487452146496416253723123, 6.86560675703035315582603165055, 7.35016128077794952947756445393, 7.52792909449380972553292741129, 8.200386287956119303385165045666, 8.292561034955820572303942187335

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