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Properties

Label 4-2592e2-1.1-c1e2-0-18
Degree $4$
Conductor $6718464$
Sign $1$
Analytic cond. $428.375$
Root an. cond. $4.54942$
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 + 4·17-s + 5·25-s + 10·29-s − 4·37-s − 10·41-s + 7·49-s + 28·53-s + 10·61-s − 12·65-s − 12·73-s + 8·85-s + 20·89-s − 18·97-s + 2·101-s + 12·109-s + 14·113-s + 11·121-s + 22·125-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 + 0.970·17-s + 25-s + 1.85·29-s − 0.657·37-s − 1.56·41-s + 49-s + 3.84·53-s + 1.28·61-s − 1.48·65-s − 1.40·73-s + 0.867·85-s + 2.11·89-s − 1.82·97-s + 0.199·101-s + 1.14·109-s + 1.31·113-s + 121-s + 1.96·125-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

\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6718464\)    =    \(2^{10} \cdot 3^{8}\)
Sign: $1$
Analytic conductor: \(428.375\)
Root analytic conductor: \(4.54942\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6718464,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.055638141\)
\(L(\frac12)\) \(\approx\) \(3.055638141\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.5.ac_ab
7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.7.a_ah
11$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.11.a_al
13$C_2^2$ \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.13.g_x
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.17.ae_bm
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.23.a_ax
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.29.ak_ct
31$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.31.a_abf
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.41.k_ch
43$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.43.a_abr
47$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.47.a_abv
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.59.a_ach
61$C_2^2$ \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) 2.61.ak_bn
67$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.67.a_acp
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.73.m_ha
79$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.79.a_adb
83$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \) 2.83.a_adf
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) 2.97.s_it
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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

−8.834070438261762304652540793764, −8.744459646820643702449843284656, −8.520443249114953747784839538589, −7.941724966536288114315934022714, −7.31353685290245761884017379474, −7.24573841553135259337695054122, −6.83690249519887679754324698026, −6.47896843890878725352938298492, −5.74753818253531788409337612762, −5.71103980586354661400137151270, −5.02106346293686316784573436189, −5.00184198945169374115743981709, −4.39220499847601658592016426977, −3.87641358932578115040214780271, −3.24902404234064968970224200129, −2.88575589535561264730649272834, −2.26182886860895408517592546414, −2.09528771784926336716786789950, −1.16144321176864410002697463006, −0.63251589691096042975129502053, 0.63251589691096042975129502053, 1.16144321176864410002697463006, 2.09528771784926336716786789950, 2.26182886860895408517592546414, 2.88575589535561264730649272834, 3.24902404234064968970224200129, 3.87641358932578115040214780271, 4.39220499847601658592016426977, 5.00184198945169374115743981709, 5.02106346293686316784573436189, 5.71103980586354661400137151270, 5.74753818253531788409337612762, 6.47896843890878725352938298492, 6.83690249519887679754324698026, 7.24573841553135259337695054122, 7.31353685290245761884017379474, 7.941724966536288114315934022714, 8.520443249114953747784839538589, 8.744459646820643702449843284656, 8.834070438261762304652540793764

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