Properties

Label 4-4608e2-1.1-c1e2-0-16
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·13-s − 12·23-s + 2·25-s − 4·37-s − 20·47-s + 12·49-s + 24·59-s + 20·61-s − 4·71-s − 20·73-s + 8·83-s + 24·97-s − 16·107-s − 20·109-s − 22·121-s + ⋯
L(s)  = 1  + 1.10·13-s − 2.50·23-s + 2/5·25-s − 0.657·37-s − 2.91·47-s + 12/7·49-s + 3.12·59-s + 2.56·61-s − 0.474·71-s − 2.34·73-s + 0.878·83-s + 2.43·97-s − 1.54·107-s − 1.91·109-s − 2·121-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: \(0\)
Selberg data: \((4,\ 21233664,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.074188180\)
\(L(\frac12)\) \(\approx\) \(2.074188180\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.7.a_am
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.13.ae_be
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \) 2.17.a_abg
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.31.a_bk
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.37.e_da
41$C_2^2$ \( 1 - 64 T^{2} + p^{2} T^{4} \) 2.41.a_acm
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.43.a_ao
47$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.47.u_hm
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.67.a_ady
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.71.e_fq
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.79.a_e
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.83.ai_ha
89$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.89.a_adc
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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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

−8.514279705462625924714808563202, −8.063185903273047506396163996512, −7.973585789856338936828036925021, −7.44099731351353017055810124025, −6.93192915184719534918215991765, −6.75383540289388016043689569726, −6.16362705844462686329400091128, −6.15975675213146004532948389346, −5.43425506182866440690025483124, −5.34930459758865093167425798010, −4.88717081131835176986691911032, −4.11834416624773577878285300205, −3.95453226816894290730954599198, −3.79972161800046962695423851304, −3.10480788238594224027819840609, −2.70818312196148287393452785584, −1.96826483377392857615872328186, −1.86607018937731840724692738260, −1.08652884750568271833373000645, −0.43000508201211822424306888142, 0.43000508201211822424306888142, 1.08652884750568271833373000645, 1.86607018937731840724692738260, 1.96826483377392857615872328186, 2.70818312196148287393452785584, 3.10480788238594224027819840609, 3.79972161800046962695423851304, 3.95453226816894290730954599198, 4.11834416624773577878285300205, 4.88717081131835176986691911032, 5.34930459758865093167425798010, 5.43425506182866440690025483124, 6.15975675213146004532948389346, 6.16362705844462686329400091128, 6.75383540289388016043689569726, 6.93192915184719534918215991765, 7.44099731351353017055810124025, 7.973585789856338936828036925021, 8.063185903273047506396163996512, 8.514279705462625924714808563202

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