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Properties

Label 4-6480e2-1.1-c1e2-0-16
Degree $4$
Conductor $41990400$
Sign $1$
Analytic cond. $2677.34$
Root an. cond. $7.19326$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·7-s + 4·11-s − 4·13-s − 8·17-s + 4·19-s + 4·23-s + 3·25-s − 10·29-s + 4·31-s − 8·35-s − 6·41-s + 16·43-s + 4·47-s + 49-s − 12·53-s + 8·55-s + 8·59-s − 10·61-s − 8·65-s − 16·67-s + 12·71-s − 16·77-s − 24·79-s − 16·83-s − 16·85-s + 6·89-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.51·7-s + 1.20·11-s − 1.10·13-s − 1.94·17-s + 0.917·19-s + 0.834·23-s + 3/5·25-s − 1.85·29-s + 0.718·31-s − 1.35·35-s − 0.937·41-s + 2.43·43-s + 0.583·47-s + 1/7·49-s − 1.64·53-s + 1.07·55-s + 1.04·59-s − 1.28·61-s − 0.992·65-s − 1.95·67-s + 1.42·71-s − 1.82·77-s − 2.70·79-s − 1.75·83-s − 1.73·85-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41990400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(41990400\)    =    \(2^{8} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2677.34\)
Root analytic conductor: \(7.19326\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 41990400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.7.e_p
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_o
13$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_s
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.17.i_bm
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$D_{4}$ \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_bv
29$C_2^2$ \( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.29.k_ct
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.37.a_abi
41$D_{4}$ \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.41.g_br
43$D_{4}$ \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.43.aq_fi
47$D_{4}$ \( 1 - 4 T + 95 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.47.ae_dr
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.53.m_fm
59$D_{4}$ \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_ba
61$D_{4}$ \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) 2.61.k_ff
67$D_{4}$ \( 1 + 16 T + 195 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.67.q_hn
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.71.am_gk
73$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.73.a_du
79$D_{4}$ \( 1 + 24 T + 290 T^{2} + 24 p T^{3} + p^{2} T^{4} \) 2.79.y_le
83$D_{4}$ \( 1 + 16 T + 203 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.83.q_hv
89$D_{4}$ \( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} \) 2.89.ag_fj
97$D_{4}$ \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_fu
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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

−7.64368664571889138734936919552, −7.35540061893367213706621863239, −6.97358340369057129050425088348, −6.93243661603426553001860792585, −6.31996445392924468166111863734, −6.11600546646132684633593309793, −5.95468938784116343163096009056, −5.35375660571117466212694220246, −4.97260508288696297677703720916, −4.65107677838850356625072135231, −4.09022210743652187098861831982, −3.91554265887523293236633708871, −3.29761567811302235412314330432, −2.93517491177492767091755272479, −2.47458382805681340335179803783, −2.29465623888969538217371935551, −1.33616080835037379873265127108, −1.33350121544644839347723045257, 0, 0, 1.33350121544644839347723045257, 1.33616080835037379873265127108, 2.29465623888969538217371935551, 2.47458382805681340335179803783, 2.93517491177492767091755272479, 3.29761567811302235412314330432, 3.91554265887523293236633708871, 4.09022210743652187098861831982, 4.65107677838850356625072135231, 4.97260508288696297677703720916, 5.35375660571117466212694220246, 5.95468938784116343163096009056, 6.11600546646132684633593309793, 6.31996445392924468166111863734, 6.93243661603426553001860792585, 6.97358340369057129050425088348, 7.35540061893367213706621863239, 7.64368664571889138734936919552

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