Properties

Label 4-4608e2-1.1-c1e2-0-59
Degree $4$
Conductor $21233664$
Sign $1$
Analytic cond. $1353.87$
Root an. cond. $6.06589$
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  + 4·5-s − 4·11-s − 4·13-s − 4·17-s − 8·23-s + 4·25-s + 12·29-s − 12·37-s + 4·41-s − 8·43-s − 8·47-s − 12·49-s + 12·53-s − 16·55-s − 8·59-s − 12·61-s − 16·65-s − 32·79-s − 12·83-s − 16·85-s + 12·89-s − 4·97-s + 4·101-s − 16·103-s + 8·107-s − 20·109-s + 4·113-s + ⋯
L(s)  = 1  + 1.78·5-s − 1.20·11-s − 1.10·13-s − 0.970·17-s − 1.66·23-s + 4/5·25-s + 2.22·29-s − 1.97·37-s + 0.624·41-s − 1.21·43-s − 1.16·47-s − 1.71·49-s + 1.64·53-s − 2.15·55-s − 1.04·59-s − 1.53·61-s − 1.98·65-s − 3.60·79-s − 1.31·83-s − 1.73·85-s + 1.27·89-s − 0.406·97-s + 0.398·101-s − 1.57·103-s + 0.773·107-s − 1.91·109-s + 0.376·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21233664\)    =    \(2^{18} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(1353.87\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.5.ae_m
7$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.7.a_m
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_s
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.13.e_w
17$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.17.e_be
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.19.a_be
23$D_{4}$ \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.23.i_cc
29$D_{4}$ \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.29.am_do
31$C_2^2$ \( 1 + 12 T^{2} + p^{2} T^{4} \) 2.31.a_m
37$D_{4}$ \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.37.m_da
41$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.41.ae_o
43$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.43.i_dq
47$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.47.i_bm
53$D_{4}$ \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.53.am_eu
59$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) 2.59.i_dy
61$D_{4}$ \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.61.m_ew
67$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.67.a_dy
71$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.71.a_cs
73$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.73.a_s
79$D_{4}$ \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \) 2.79.bg_pw
83$D_{4}$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.83.m_hm
89$C_4$ \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) 2.89.am_di
97$D_{4}$ \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.97.e_gk
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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

−8.275118449881244459627684166650, −7.70590967242162271999165377708, −7.50749029383517847795444444746, −6.91534987817102164710370500071, −6.52668567758684681507343657465, −6.46533773012563798100921202172, −5.79611135750805509784797974983, −5.70788758671310501615544982912, −5.09745297923845429809782459322, −5.01556975539771768558148011217, −4.37442042352862026103270288957, −4.27400599758714745701000232862, −3.33759649628323766866561039116, −2.99817547786893384080430872515, −2.48273669960225229936740453600, −2.29918528780258414554203302300, −1.57646157098888365968633933577, −1.53907765501447248381222267162, 0, 0, 1.53907765501447248381222267162, 1.57646157098888365968633933577, 2.29918528780258414554203302300, 2.48273669960225229936740453600, 2.99817547786893384080430872515, 3.33759649628323766866561039116, 4.27400599758714745701000232862, 4.37442042352862026103270288957, 5.01556975539771768558148011217, 5.09745297923845429809782459322, 5.70788758671310501615544982912, 5.79611135750805509784797974983, 6.46533773012563798100921202172, 6.52668567758684681507343657465, 6.91534987817102164710370500071, 7.50749029383517847795444444746, 7.70590967242162271999165377708, 8.275118449881244459627684166650

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