Properties

Label 4-9200e2-1.1-c1e2-0-3
Degree $4$
Conductor $84640000$
Sign $1$
Analytic cond. $5396.71$
Root an. cond. $8.57101$
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  + 3-s + 3·7-s − 4·9-s + 9·11-s − 3·13-s + 5·17-s + 7·19-s + 3·21-s + 2·23-s − 6·27-s − 6·29-s + 11·31-s + 9·33-s + 6·37-s − 3·39-s − 9·41-s + 18·43-s − 10·47-s + 4·49-s + 5·51-s + 4·53-s + 7·57-s + 12·59-s − 61-s − 12·63-s + 6·67-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 4/3·9-s + 2.71·11-s − 0.832·13-s + 1.21·17-s + 1.60·19-s + 0.654·21-s + 0.417·23-s − 1.15·27-s − 1.11·29-s + 1.97·31-s + 1.56·33-s + 0.986·37-s − 0.480·39-s − 1.40·41-s + 2.74·43-s − 1.45·47-s + 4/7·49-s + 0.700·51-s + 0.549·53-s + 0.927·57-s + 1.56·59-s − 0.128·61-s − 1.51·63-s + 0.733·67-s + 0.240·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84640000\)    =    \(2^{8} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(5396.71\)
Root analytic conductor: \(8.57101\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 84640000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.426655274\)
\(L(\frac12)\) \(\approx\) \(7.426655274\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) 2.3.ab_f
7$D_{4}$ \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.7.ad_f
11$C_4$ \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.11.aj_bp
13$D_{4}$ \( 1 + 3 T + 27 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.13.d_bb
17$D_{4}$ \( 1 - 5 T + 29 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.17.af_bd
19$C_4$ \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \) 2.19.ah_bn
29$D_{4}$ \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.29.g_w
31$C_4$ \( 1 - 11 T + 81 T^{2} - 11 p T^{3} + p^{2} T^{4} \) 2.31.al_dd
37$D_{4}$ \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.37.ag_bm
41$D_{4}$ \( 1 + 9 T + 101 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.41.j_dx
43$C_2^2$ \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.43.as_gg
47$D_{4}$ \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.47.k_cw
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.59.am_fy
61$D_{4}$ \( 1 + T + 111 T^{2} + p T^{3} + p^{2} T^{4} \) 2.61.b_eh
67$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.67.ag_fi
71$D_{4}$ \( 1 + 3 T + 143 T^{2} + 3 p T^{3} + p^{2} T^{4} \) 2.71.d_fn
73$D_{4}$ \( 1 - 24 T + 270 T^{2} - 24 p T^{3} + p^{2} T^{4} \) 2.73.ay_kk
79$D_{4}$ \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.79.ac_ek
83$D_{4}$ \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.83.ac_es
89$D_{4}$ \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.89.g_fm
97$D_{4}$ \( 1 - 3 T - 15 T^{2} - 3 p T^{3} + p^{2} T^{4} \) 2.97.ad_ap
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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

−7.86865038545985921417933507348, −7.68599197309634438857352845937, −7.13640210844700763732453290781, −7.09325830199187544717824784188, −6.38436847980069810317686416078, −6.29187798516764691942004930954, −5.83762439020066886865490495127, −5.43620966039156213947596491617, −5.13314637995467359070377944920, −4.84736731707165934214205527080, −4.38458250908403905248070270349, −3.86889534189187216597359948946, −3.58585828563339308843126071165, −3.42735965729922102484923547998, −2.65353596535418697113938178495, −2.60391050593865252686722508458, −1.96423764845044532578546539758, −1.42504954846199028369721397294, −0.978525428787221540032160992044, −0.72509351005268660818841877275, 0.72509351005268660818841877275, 0.978525428787221540032160992044, 1.42504954846199028369721397294, 1.96423764845044532578546539758, 2.60391050593865252686722508458, 2.65353596535418697113938178495, 3.42735965729922102484923547998, 3.58585828563339308843126071165, 3.86889534189187216597359948946, 4.38458250908403905248070270349, 4.84736731707165934214205527080, 5.13314637995467359070377944920, 5.43620966039156213947596491617, 5.83762439020066886865490495127, 6.29187798516764691942004930954, 6.38436847980069810317686416078, 7.09325830199187544717824784188, 7.13640210844700763732453290781, 7.68599197309634438857352845937, 7.86865038545985921417933507348

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