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Properties

Label 4-5808e2-1.1-c1e2-0-21
Degree $4$
Conductor $33732864$
Sign $1$
Analytic cond. $2150.83$
Root an. cond. $6.81007$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5·5-s − 7-s + 3·9-s − 2·13-s − 10·15-s − 8·17-s + 2·19-s + 2·21-s − 2·23-s + 10·25-s − 4·27-s − 7·29-s + 9·31-s − 5·35-s + 4·39-s − 2·41-s − 14·43-s + 15·45-s − 8·47-s − 12·49-s + 16·51-s + 5·53-s − 4·57-s − 5·59-s − 6·61-s − 3·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 2.23·5-s − 0.377·7-s + 9-s − 0.554·13-s − 2.58·15-s − 1.94·17-s + 0.458·19-s + 0.436·21-s − 0.417·23-s + 2·25-s − 0.769·27-s − 1.29·29-s + 1.61·31-s − 0.845·35-s + 0.640·39-s − 0.312·41-s − 2.13·43-s + 2.23·45-s − 1.16·47-s − 1.71·49-s + 2.24·51-s + 0.686·53-s − 0.529·57-s − 0.650·59-s − 0.768·61-s − 0.377·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33732864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33732864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(33732864\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(2150.83\)
Root analytic conductor: \(6.81007\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 33732864,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
11 \( 1 \)
good5$C_4$ \( 1 - p T + 3 p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) 2.5.af_p
7$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \) 2.7.b_n
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.13.c_w
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_4$ \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) 2.19.ac_ag
23$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.23.c_c
29$D_{4}$ \( 1 + 7 T + 59 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.29.h_ch
31$D_{4}$ \( 1 - 9 T + 81 T^{2} - 9 p T^{3} + p^{2} T^{4} \) 2.31.aj_dd
37$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.37.a_ag
41$D_{4}$ \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.41.c_bm
43$D_{4}$ \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.43.o_fa
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$D_{4}$ \( 1 - 5 T + 111 T^{2} - 5 p T^{3} + p^{2} T^{4} \) 2.53.af_eh
59$D_{4}$ \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) 2.59.f_cl
61$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) 2.61.g_di
67$D_{4}$ \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.67.ai_cs
71$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.71.m_gw
73$D_{4}$ \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) 2.73.j_cn
79$D_{4}$ \( 1 + 7 T + 159 T^{2} + 7 p T^{3} + p^{2} T^{4} \) 2.79.h_gd
83$D_{4}$ \( 1 - T + 165 T^{2} - p T^{3} + p^{2} T^{4} \) 2.83.ab_gj
89$D_{4}$ \( 1 - 18 T + 254 T^{2} - 18 p T^{3} + p^{2} T^{4} \) 2.89.as_ju
97$D_{4}$ \( 1 + 17 T + 265 T^{2} + 17 p T^{3} + p^{2} T^{4} \) 2.97.r_kf
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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

−7.79562232960162737129204953049, −7.58400938502567867126198090964, −6.81796850459674994596658656618, −6.75572814830645800903665282147, −6.36835009059892470187141808637, −6.33589701273058871895916771160, −5.67870086829817110376636649529, −5.61557082349695976731923216005, −4.99650691482567064862945703604, −4.94397751157220243051932820819, −4.39725683810943817362758123210, −4.09034561751467517470173784597, −3.26633220273143037365761321221, −3.03234233075914286947968510606, −2.26808276515059831874415750002, −2.12818914080327731776223029574, −1.54823336130426341026835301586, −1.29696886923807264871057922779, 0, 0, 1.29696886923807264871057922779, 1.54823336130426341026835301586, 2.12818914080327731776223029574, 2.26808276515059831874415750002, 3.03234233075914286947968510606, 3.26633220273143037365761321221, 4.09034561751467517470173784597, 4.39725683810943817362758123210, 4.94397751157220243051932820819, 4.99650691482567064862945703604, 5.61557082349695976731923216005, 5.67870086829817110376636649529, 6.33589701273058871895916771160, 6.36835009059892470187141808637, 6.75572814830645800903665282147, 6.81796850459674994596658656618, 7.58400938502567867126198090964, 7.79562232960162737129204953049

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