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Properties

Label 4-623808-1.1-c1e2-0-65
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s − 4-s − 6-s + 4·7-s + 3·8-s + 9-s − 12-s − 4·14-s − 16-s − 18-s − 4·19-s + 4·21-s + 3·24-s − 10·25-s + 27-s − 4·28-s − 8·29-s − 5·32-s − 36-s + 4·38-s − 8·41-s − 4·42-s − 8·43-s − 48-s + 2·49-s + 10·50-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.288·12-s − 1.06·14-s − 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.872·21-s + 0.612·24-s − 2·25-s + 0.192·27-s − 0.755·28-s − 1.48·29-s − 0.883·32-s − 1/6·36-s + 0.648·38-s − 1.24·41-s − 0.617·42-s − 1.21·43-s − 0.144·48-s + 2/7·49-s + 1.41·50-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: \(1\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 + 4 T + p T^{2} \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.7.ae_o
11$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.11.a_o
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
23$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.23.a_c
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.i_cs
31$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.31.a_by
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.i_ck
43$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.i_di
47$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.47.a_c
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.53.ai_eo
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.a_aba
61$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.61.aq_fu
67$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.67.a_abq
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$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.am_eo
79$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.79.a_c
83$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.83.a_as
89$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.i_dq
97$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.97.a_ade
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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

−8.193695355455962012461491792334, −7.897179848666337609198487573404, −7.56411870664652821582677747215, −6.97230499077490902226536338024, −6.55119130341689932145439434740, −5.71969138736606233742216533202, −5.27555252851420954848695452535, −4.96465004741004469032539347110, −4.21410916106122281136368301439, −3.95018849599060273532505664097, −3.44397170827507297731826938859, −2.24135760861622127054082439094, −1.94961424928317279852299617913, −1.32639999422257296683032682274, 0, 1.32639999422257296683032682274, 1.94961424928317279852299617913, 2.24135760861622127054082439094, 3.44397170827507297731826938859, 3.95018849599060273532505664097, 4.21410916106122281136368301439, 4.96465004741004469032539347110, 5.27555252851420954848695452535, 5.71969138736606233742216533202, 6.55119130341689932145439434740, 6.97230499077490902226536338024, 7.56411870664652821582677747215, 7.897179848666337609198487573404, 8.193695355455962012461491792334

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