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Properties

Label 4-72e4-1.1-c1e2-0-6
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.c_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.ak_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.k_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.e_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.e_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.aba_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

−8.150917327539827677008778213859, −7.931359034080329901453567671558, −7.67571514345711865054333088078, −7.42033465512105785760451530098, −7.19085507708317891836448794155, −6.69268498541615455265485688743, −6.18946316145061602789721299054, −5.70112884439613088567484230552, −5.50491567628131367817655087555, −5.18750725072715391675467553256, −4.79975563683551527017990764773, −4.12718724321880396199702035125, −3.83897759561025808591519441554, −3.73708546635097583592927245371, −3.00514277623684474839794654782, −2.77505797384630348125204967216, −1.94672129062779631945298853098, −1.90292715359178335365249662513, −0.797634701657125513870414549399, −0.50875853629951289157107948415, 0.50875853629951289157107948415, 0.797634701657125513870414549399, 1.90292715359178335365249662513, 1.94672129062779631945298853098, 2.77505797384630348125204967216, 3.00514277623684474839794654782, 3.73708546635097583592927245371, 3.83897759561025808591519441554, 4.12718724321880396199702035125, 4.79975563683551527017990764773, 5.18750725072715391675467553256, 5.50491567628131367817655087555, 5.70112884439613088567484230552, 6.18946316145061602789721299054, 6.69268498541615455265485688743, 7.19085507708317891836448794155, 7.42033465512105785760451530098, 7.67571514345711865054333088078, 7.931359034080329901453567671558, 8.150917327539827677008778213859

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