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Properties

Label 4-4160e2-1.1-c1e2-0-4
Degree $4$
Conductor $17305600$
Sign $1$
Analytic cond. $1103.42$
Root an. cond. $5.76348$
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 + 4·7-s + 4·11-s + 2·13-s + 4·17-s + 4·19-s + 8·23-s + 3·25-s − 8·29-s − 4·31-s − 8·35-s + 12·41-s − 8·43-s − 4·47-s − 2·49-s − 12·53-s − 8·55-s − 4·59-s − 8·61-s − 4·65-s + 20·67-s − 12·71-s + 16·77-s − 9·81-s − 4·83-s − 8·85-s + 12·89-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s + 0.970·17-s + 0.917·19-s + 1.66·23-s + 3/5·25-s − 1.48·29-s − 0.718·31-s − 1.35·35-s + 1.87·41-s − 1.21·43-s − 0.583·47-s − 2/7·49-s − 1.64·53-s − 1.07·55-s − 0.520·59-s − 1.02·61-s − 0.496·65-s + 2.44·67-s − 1.42·71-s + 1.82·77-s − 81-s − 0.439·83-s − 0.867·85-s + 1.27·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17305600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17305600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17305600\)    =    \(2^{12} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1103.42\)
Root analytic conductor: \(5.76348\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 17305600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.720013264\)
\(L(\frac12)\) \(\approx\) \(3.720013264\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 + p^{2} T^{4} \) 2.3.a_a
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.7.ae_s
11$D_{4}$ \( 1 - 4 T + 20 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.11.ae_u
17$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.17.ae_o
19$D_{4}$ \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_bk
23$D_{4}$ \( 1 - 8 T + 56 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.23.ai_ce
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.29.i_cw
31$D_{4}$ \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.31.e_ci
37$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.37.a_by
41$D_{4}$ \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.41.am_dq
43$D_{4}$ \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_bw
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.47.e_du
53$D_{4}$ \( 1 + 12 T + 118 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.53.m_eo
59$D_{4}$ \( 1 + 4 T + 68 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.59.e_cq
61$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.61.i_bq
67$D_{4}$ \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) 2.67.au_ic
71$D_{4}$ \( 1 + 12 T + 172 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.71.m_gq
73$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.73.a_acs
79$C_2^2$ \( 1 + 134 T^{2} + p^{2} T^{4} \) 2.79.a_fe
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.83.e_go
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$D_{4}$ \( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_dy
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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

−8.536979992933136069643445024711, −8.241931850839087000690124940171, −7.63125649651268229857789424226, −7.62451251039985565438992329163, −7.39649727999998275308944915334, −6.85735996163673941950735150808, −6.33232698311946482134425396355, −6.14817539055921566747033333234, −5.38410548459870093398866412604, −5.32613120825045798844904726491, −4.82130743277414439781138857869, −4.44823954608155168108917027369, −4.11252064825140706616281753788, −3.50303263945378926451702693452, −3.26058460697073078745467769467, −2.99333662394091491053422040047, −1.91509117466914696146193422396, −1.65836754373225856382646726732, −1.17875630867796752775651763657, −0.62069008302814059037132102165, 0.62069008302814059037132102165, 1.17875630867796752775651763657, 1.65836754373225856382646726732, 1.91509117466914696146193422396, 2.99333662394091491053422040047, 3.26058460697073078745467769467, 3.50303263945378926451702693452, 4.11252064825140706616281753788, 4.44823954608155168108917027369, 4.82130743277414439781138857869, 5.32613120825045798844904726491, 5.38410548459870093398866412604, 6.14817539055921566747033333234, 6.33232698311946482134425396355, 6.85735996163673941950735150808, 7.39649727999998275308944915334, 7.62451251039985565438992329163, 7.63125649651268229857789424226, 8.241931850839087000690124940171, 8.536979992933136069643445024711

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