Properties

Label 4-4608e2-1.1-c1e2-0-21
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·5-s − 10·13-s − 4·17-s + 18·25-s + 14·29-s − 14·37-s + 14·49-s + 10·53-s − 22·61-s + 60·65-s + 24·85-s + 16·97-s − 22·101-s + 26·109-s + 32·113-s − 30·125-s + ⋯
L(s)  = 1  − 2.68·5-s − 2.77·13-s − 0.970·17-s + 18/5·25-s + 2.59·29-s − 2.30·37-s + 2·49-s + 1.37·53-s − 2.81·61-s + 7.44·65-s + 2.60·85-s + 1.62·97-s − 2.18·101-s + 2.49·109-s + 3.01·113-s − 2.68·125-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

\(L(1)\) \(\approx\) \(0.7867378809\)
\(L(\frac12)\) \(\approx\) \(0.7867378809\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.g_s
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.17.e_bm
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.29.ao_du
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.o_du
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ak_by
59$C_2^2$ \( 1 + p^{2} T^{4} \) 2.59.a_a
61$C_2$ \( ( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.w_ji
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.71.a_afm
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.79.a_gc
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.89.a_da
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.97.aq_jy
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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

−8.399062568048950304584489988199, −8.166195734400201084197002231667, −7.57567767114120903373469870029, −7.45269951706732397188339575250, −7.16143772997174155151421621112, −7.01006313628404228510543896245, −6.48061008979165259843088466333, −6.05717688800115152783772181445, −5.29295450114054548050252992470, −5.04423156442628978996083413139, −4.64444951772581259008504095291, −4.38021264869834573793164796090, −4.16333989472650239450548716520, −3.59162463140568674735929736646, −3.07041713110177804926037250965, −2.85368016125970754574402632592, −2.31040179454647455887921419733, −1.75043298355863877128096070713, −0.53015996554517306320105038822, −0.48526667414940587755408887440, 0.48526667414940587755408887440, 0.53015996554517306320105038822, 1.75043298355863877128096070713, 2.31040179454647455887921419733, 2.85368016125970754574402632592, 3.07041713110177804926037250965, 3.59162463140568674735929736646, 4.16333989472650239450548716520, 4.38021264869834573793164796090, 4.64444951772581259008504095291, 5.04423156442628978996083413139, 5.29295450114054548050252992470, 6.05717688800115152783772181445, 6.48061008979165259843088466333, 7.01006313628404228510543896245, 7.16143772997174155151421621112, 7.45269951706732397188339575250, 7.57567767114120903373469870029, 8.166195734400201084197002231667, 8.399062568048950304584489988199

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