↑

Properties

Label 4-2268e2-1.1-c1e2-0-32
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 2·11-s + 10·17-s + 4·19-s − 2·23-s + 5·25-s − 10·29-s − 35-s + 10·37-s − 3·41-s + 7·43-s + 3·47-s + 12·53-s + 2·55-s − 59-s + 6·61-s − 4·67-s + 16·71-s + 20·73-s − 2·77-s + 3·79-s + 13·83-s − 10·85-s − 12·89-s − 4·95-s + 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.603·11-s + 2.42·17-s + 0.917·19-s − 0.417·23-s + 25-s − 1.85·29-s − 0.169·35-s + 1.64·37-s − 0.468·41-s + 1.06·43-s + 0.437·47-s + 1.64·53-s + 0.269·55-s − 0.130·59-s + 0.768·61-s − 0.488·67-s + 1.89·71-s + 2.34·73-s − 0.227·77-s + 0.337·79-s + 1.42·83-s − 1.08·85-s − 1.27·89-s − 0.410·95-s + 1.42·97-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

\(L(1)\) \(\approx\) \(2.911745990\)
\(L(\frac12)\) \(\approx\) \(2.911745990\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + T^{2} \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) 2.5.b_ae
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.13.a_an
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_at
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_ct
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.37.ak_dv
41$C_2^2$ \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.41.d_abg
43$C_2^2$ \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.43.ah_g
47$C_2^2$ \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.47.ad_abm
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2^2$ \( 1 + T - 58 T^{2} + p T^{3} + p^{2} T^{4} \) 2.59.b_acg
61$C_2^2$ \( 1 - 6 T - 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.61.ag_az
67$C_2^2$ \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.67.e_abz
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.71.aq_hy
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.73.au_jm
79$C_2^2$ \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.79.ad_acs
83$C_2^2$ \( 1 - 13 T + 86 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.83.an_di
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.97.ao_dv
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.260064518203329531618875526483, −8.915740154131818643915087986886, −8.097206928770103965389767492822, −8.013279859638913198538110714825, −7.83572905277717634772146245286, −7.46610698979839689644916567187, −6.82159292411992422863343958801, −6.75630883234863414172488438950, −5.81537288188626637384075543930, −5.60171153629481598290722677962, −5.36464691720324482658378196311, −5.01655764935268286998097405800, −4.15371081928687495788610105494, −4.07460364697784263766479694543, −3.29798830095274976485712066235, −3.20722069943381916092981797082, −2.42784810832185561248659697142, −1.96782380358481909452475251995, −1.02368567540450451274638806551, −0.74362518436616863624003236738, 0.74362518436616863624003236738, 1.02368567540450451274638806551, 1.96782380358481909452475251995, 2.42784810832185561248659697142, 3.20722069943381916092981797082, 3.29798830095274976485712066235, 4.07460364697784263766479694543, 4.15371081928687495788610105494, 5.01655764935268286998097405800, 5.36464691720324482658378196311, 5.60171153629481598290722677962, 5.81537288188626637384075543930, 6.75630883234863414172488438950, 6.82159292411992422863343958801, 7.46610698979839689644916567187, 7.83572905277717634772146245286, 8.013279859638913198538110714825, 8.097206928770103965389767492822, 8.915740154131818643915087986886, 9.260064518203329531618875526483

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