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Properties

Label 4-3744e2-1.1-c1e2-0-6
Degree $4$
Conductor $14017536$
Sign $1$
Analytic cond. $893.770$
Root an. cond. $5.46772$
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 + 10·17-s − 12·23-s + 25-s + 8·29-s + 6·43-s + 13·49-s + 4·53-s − 24·61-s − 12·79-s + 8·101-s + 12·103-s + 24·107-s + 28·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1.10·13-s + 2.42·17-s − 2.50·23-s + 1/5·25-s + 1.48·29-s + 0.914·43-s + 13/7·49-s + 0.549·53-s − 3.07·61-s − 1.35·79-s + 0.796·101-s + 1.18·103-s + 2.32·107-s + 2.63·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14017536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14017536\)    =    \(2^{10} \cdot 3^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(893.770\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14017536,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.536969275\)
\(L(\frac12)\) \(\approx\) \(2.536969275\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.5.a_ab
7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
17$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.17.ak_ch
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.23.m_de
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.29.ai_cw
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.37.a_acn
41$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.41.a_ck
43$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.43.ag_dr
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.53.ae_eg
59$C_2^2$ \( 1 - 114 T^{2} + p^{2} T^{4} \) 2.59.a_aek
61$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.61.y_kg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.71.a_av
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.a_aeg
79$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.79.m_hm
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−8.472414656446791305406028022608, −8.414338324498168263162985059773, −7.88477837267182302627265850852, −7.49376815407816723934314199972, −7.41734869520226527993282390790, −7.04853892827756021920271825999, −6.29084651236932790981597008415, −6.06151937086332254916236812499, −5.62662236167063509197229071273, −5.61106895281596886499308642303, −4.76132448633334948552225021401, −4.55607388916724377240193859330, −4.20012580939356378941882626180, −3.61098628760304238628394547084, −3.05221680078805786128629294448, −2.98091515633812840203230871740, −2.08739496162062234136425520890, −1.92502279841706985992385189298, −1.02652043938086240876879497261, −0.54055482276740878911031910451, 0.54055482276740878911031910451, 1.02652043938086240876879497261, 1.92502279841706985992385189298, 2.08739496162062234136425520890, 2.98091515633812840203230871740, 3.05221680078805786128629294448, 3.61098628760304238628394547084, 4.20012580939356378941882626180, 4.55607388916724377240193859330, 4.76132448633334948552225021401, 5.61106895281596886499308642303, 5.62662236167063509197229071273, 6.06151937086332254916236812499, 6.29084651236932790981597008415, 7.04853892827756021920271825999, 7.41734869520226527993282390790, 7.49376815407816723934314199972, 7.88477837267182302627265850852, 8.414338324498168263162985059773, 8.472414656446791305406028022608

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