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Properties

Label 4-623808-1.1-c1e2-0-29
Degree $4$
Conductor $623808$
Sign $1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s + 16-s − 18-s + 2·19-s − 24-s + 2·25-s + 27-s + 14·29-s − 32-s + 36-s − 2·38-s + 14·41-s + 4·43-s + 48-s − 10·49-s − 2·50-s − 10·53-s − 54-s + 2·57-s − 14·58-s + 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 1/4·16-s − 0.235·18-s + 0.458·19-s − 0.204·24-s + 2/5·25-s + 0.192·27-s + 2.59·29-s − 0.176·32-s + 1/6·36-s − 0.324·38-s + 2.18·41-s + 0.609·43-s + 0.144·48-s − 1.42·49-s − 0.282·50-s − 1.37·53-s − 0.136·54-s + 0.264·57-s − 1.83·58-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 623808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(623808\)    =    \(2^{6} \cdot 3^{3} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(39.7745\)
Root analytic conductor: \(2.51131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.896882838\)
\(L(\frac12)\) \(\approx\) \(1.896882838\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.13.a_ao
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
23$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.23.a_ao
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.29.ao_ec
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.31.a_s
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.41.ao_fa
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.43.ae_cc
47$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.47.a_acc
53$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.k_ec
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.am_eo
61$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.61.ae_eg
67$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.67.a_ack
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.73.q_hy
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.79.a_ac
83$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.83.a_s
89$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.c_abu
97$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.97.a_ack
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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

−8.464285392266424663929213439399, −7.986237060436625361103869596655, −7.64084036516900287560552417530, −7.07856249620893156480448347167, −6.78186504535928796675893061600, −6.08213828605181351171017922528, −5.93913423463641034444986857125, −4.97413152367818321297995387876, −4.70373187977964433046919900150, −4.06463866818455884200497669592, −3.39070489301162048479309034347, −2.76952847741667438604206140239, −2.48179111345703279268915095192, −1.48699070939147802447318199330, −0.812856957188175653390279653740, 0.812856957188175653390279653740, 1.48699070939147802447318199330, 2.48179111345703279268915095192, 2.76952847741667438604206140239, 3.39070489301162048479309034347, 4.06463866818455884200497669592, 4.70373187977964433046919900150, 4.97413152367818321297995387876, 5.93913423463641034444986857125, 6.08213828605181351171017922528, 6.78186504535928796675893061600, 7.07856249620893156480448347167, 7.64084036516900287560552417530, 7.986237060436625361103869596655, 8.464285392266424663929213439399

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