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Properties

Label 4-2268e2-1.1-c1e2-0-31
Degree $4$
Conductor $5143824$
Sign $1$
Analytic cond. $327.974$
Root an. cond. $4.25559$
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  + 7-s + 3·13-s + 5·25-s + 18·31-s + 2·37-s + 8·43-s − 6·49-s − 15·61-s − 11·67-s + 13·79-s + 3·91-s + 33·97-s + 33·103-s − 4·109-s − 11·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7·169-s + 173-s + 5·175-s + ⋯
L(s)  = 1  + 0.377·7-s + 0.832·13-s + 25-s + 3.23·31-s + 0.328·37-s + 1.21·43-s − 6/7·49-s − 1.92·61-s − 1.34·67-s + 1.46·79-s + 0.314·91-s + 3.35·97-s + 3.25·103-s − 0.383·109-s − 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 − 0.538·169-s + 0.0760·173-s + 0.377·175-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5143824\)    =    \(2^{4} \cdot 3^{8} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(327.974\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5143824,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.243349688\)
\(L(\frac12)\) \(\approx\) \(3.243349688\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - T + p T^{2} \)
good5$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.5.a_af
11$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.11.a_l
13$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.ad_q
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) 2.19.a_bl
23$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.23.a_x
29$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.29.a_bd
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.31.as_fj
37$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.37.ac_cx
41$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.41.a_abp
43$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.ai_v
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.53.a_aec
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2$ \( ( 1 + T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.p_fg
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.67.l_cc
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) 2.73.a_afn
79$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.an_dm
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 - 14 T + p T^{2} ) \) 2.97.abh_rs
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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

−8.957845594041777441685976503606, −8.816461954399482816526542109352, −8.614880188907827938813873798234, −7.977926946483047116697374987659, −7.60180260246159955692598488342, −7.58884737148203376123949510851, −6.74191425909987842746220278727, −6.45475002832406033965570038753, −6.05344493236716187471932821006, −5.95103662025693935911652799337, −5.07074140562705290133021837929, −4.77873509028592972501557530030, −4.50837833703573721072753568677, −4.04414834191013381453118875332, −3.23142853138586706696554153105, −3.14281811639296592433677437549, −2.46893187669859520433949196678, −1.90042347101419496517922453147, −1.13158635465035578929384526109, −0.73437428858594037551216184151, 0.73437428858594037551216184151, 1.13158635465035578929384526109, 1.90042347101419496517922453147, 2.46893187669859520433949196678, 3.14281811639296592433677437549, 3.23142853138586706696554153105, 4.04414834191013381453118875332, 4.50837833703573721072753568677, 4.77873509028592972501557530030, 5.07074140562705290133021837929, 5.95103662025693935911652799337, 6.05344493236716187471932821006, 6.45475002832406033965570038753, 6.74191425909987842746220278727, 7.58884737148203376123949510851, 7.60180260246159955692598488342, 7.977926946483047116697374987659, 8.614880188907827938813873798234, 8.816461954399482816526542109352, 8.957845594041777441685976503606

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