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Properties

Label 4-623808-1.1-c1e2-0-17
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 + 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: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836792309\)
\(L(\frac12)\) \(\approx\) \(1.836792309\)
\(L(\frac{3}{2})\) 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$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ai_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$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ai_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$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.i_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$ \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.ai_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

−8.205305301997884072308827756544, −8.025041166902737359381279506537, −7.66967227920303173869151788138, −6.83175720670221093907505277815, −6.42937330798760142767942402354, −6.06322540585740643749187395617, −5.40810087233288337245655766610, −5.16481935274214314729936395501, −4.63019587562806993069134994256, −4.26008544690123010863342268862, −3.84645169544047642440451115445, −3.11689737132359397540898951388, −2.25168271508361285407176222423, −1.71780578872879968822343550729, −0.64713370889969897804292920038, 0.64713370889969897804292920038, 1.71780578872879968822343550729, 2.25168271508361285407176222423, 3.11689737132359397540898951388, 3.84645169544047642440451115445, 4.26008544690123010863342268862, 4.63019587562806993069134994256, 5.16481935274214314729936395501, 5.40810087233288337245655766610, 6.06322540585740643749187395617, 6.42937330798760142767942402354, 6.83175720670221093907505277815, 7.66967227920303173869151788138, 8.025041166902737359381279506537, 8.205305301997884072308827756544

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