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Properties

Label 4-623808-1.1-c1e2-0-32
Degree $4$
Conductor $623808$
Sign $1$
Analytic cond. $39.7745$
Root an. cond. $2.51131$
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  − 3-s + 8·7-s + 9-s − 8·13-s + 2·19-s − 8·21-s + 6·25-s − 27-s + 4·31-s + 8·37-s + 8·39-s + 8·43-s + 34·49-s − 2·57-s − 20·61-s + 8·63-s + 16·67-s − 4·73-s − 6·75-s + 28·79-s + 81-s − 64·91-s − 4·93-s + 28·97-s − 4·103-s − 32·109-s − 8·111-s + ⋯
L(s)  = 1  − 0.577·3-s + 3.02·7-s + 1/3·9-s − 2.21·13-s + 0.458·19-s − 1.74·21-s + 6/5·25-s − 0.192·27-s + 0.718·31-s + 1.31·37-s + 1.28·39-s + 1.21·43-s + 34/7·49-s − 0.264·57-s − 2.56·61-s + 1.00·63-s + 1.95·67-s − 0.468·73-s − 0.692·75-s + 3.15·79-s + 1/9·81-s − 6.70·91-s − 0.414·93-s + 2.84·97-s − 0.394·103-s − 3.06·109-s − 0.759·111-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: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 623808,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.348289350\)
\(L(\frac12)\) \(\approx\) \(2.348289350\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.7.ai_be
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.11.a_g
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.23.a_k
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.a_cc
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.37.ai_dm
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.47.a_dm
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.a_dy
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.73.e_fu
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.83.a_adm
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−8.213102607783853972181362930470, −7.83209972276749201593977871816, −7.47928872381719998670522703456, −7.39148933340994032193671942533, −6.56715591626159015914612137952, −6.08417078961404314892413228818, −5.32267004942502611581924636194, −4.98783298689795603138187378693, −4.80382850580496834483645089906, −4.50335507641114582801242921882, −3.76647440070571202785776880394, −2.58336626686734098148346747903, −2.38466653285533258222457062316, −1.53184510895126931162943066328, −0.876714004684985756350914094634, 0.876714004684985756350914094634, 1.53184510895126931162943066328, 2.38466653285533258222457062316, 2.58336626686734098148346747903, 3.76647440070571202785776880394, 4.50335507641114582801242921882, 4.80382850580496834483645089906, 4.98783298689795603138187378693, 5.32267004942502611581924636194, 6.08417078961404314892413228818, 6.56715591626159015914612137952, 7.39148933340994032193671942533, 7.47928872381719998670522703456, 7.83209972276749201593977871816, 8.213102607783853972181362930470

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