Properties

Label 4-2592e2-1.1-c1e2-0-3
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 − 7-s − 2·11-s − 13-s + 12·17-s − 10·19-s + 6·23-s + 5·25-s − 8·29-s − 8·31-s + 2·35-s − 10·37-s − 8·41-s + 4·43-s − 10·47-s + 7·49-s + 8·53-s + 4·55-s + 14·59-s − 3·61-s + 2·65-s + 13·67-s + 8·71-s + 18·73-s + 2·77-s − 11·79-s − 12·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 0.603·11-s − 0.277·13-s + 2.91·17-s − 2.29·19-s + 1.25·23-s + 25-s − 1.48·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s − 1.24·41-s + 0.609·43-s − 1.45·47-s + 49-s + 1.09·53-s + 0.539·55-s + 1.82·59-s − 0.384·61-s + 0.248·65-s + 1.58·67-s + 0.949·71-s + 2.10·73-s + 0.227·77-s − 1.23·79-s − 1.31·83-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

\(L(1)\) \(\approx\) \(1.005207930\)
\(L(\frac12)\) \(\approx\) \(1.005207930\)
\(L(\frac{3}{2})\) 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.c_ab
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.7.b_ag
11$C_2^2$ \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.11.c_ah
13$C_2^2$ \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) 2.13.b_am
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.17.am_cs
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.19.k_cl
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.23.ag_n
29$C_2^2$ \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.29.i_bj
31$C_2^2$ \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.31.i_bh
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.37.k_dv
41$C_2^2$ \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.41.i_x
43$C_2^2$ \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.43.ae_abb
47$C_2^2$ \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_cb
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.53.ai_es
59$C_2^2$ \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.59.ao_fh
61$C_2^2$ \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.61.d_aca
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) 2.67.an_dy
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \) 2.73.as_it
79$C_2^2$ \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) 2.79.l_bq
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_cj
89$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.89.e_ha
97$C_2^2$ \( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} \) 2.97.b_ads
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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

−8.983209757758252815104287735672, −8.506280981085827258676944065772, −8.346812771950802682173692420936, −8.023205247063217770380306113120, −7.53447336677812027102722253324, −7.04505635537817955425168739501, −6.96244770770211698235830948959, −6.63316497795641254071546618942, −5.62744903114759413230907644897, −5.61512121159976493635059759834, −5.34872448735191040196449811286, −4.76250329269470778886795209927, −4.24486905978201178992458255098, −3.63425248054337213612292633150, −3.48690876803487961968732657726, −3.16654151727281908286106669513, −2.33222440703395001109327315120, −1.95127430833541022792962749106, −1.12659760927974406734232360859, −0.36470484389903109025466912874, 0.36470484389903109025466912874, 1.12659760927974406734232360859, 1.95127430833541022792962749106, 2.33222440703395001109327315120, 3.16654151727281908286106669513, 3.48690876803487961968732657726, 3.63425248054337213612292633150, 4.24486905978201178992458255098, 4.76250329269470778886795209927, 5.34872448735191040196449811286, 5.61512121159976493635059759834, 5.62744903114759413230907644897, 6.63316497795641254071546618942, 6.96244770770211698235830948959, 7.04505635537817955425168739501, 7.53447336677812027102722253324, 8.023205247063217770380306113120, 8.346812771950802682173692420936, 8.506280981085827258676944065772, 8.983209757758252815104287735672

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