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Properties

Label 4-623808-1.1-c1e2-0-19
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  + 3-s − 2·4-s + 2·7-s + 9-s − 2·12-s + 4·16-s + 2·21-s − 7·25-s + 27-s − 4·28-s − 4·29-s − 2·36-s + 16·43-s + 4·48-s − 7·49-s + 24·53-s + 4·59-s − 4·61-s + 2·63-s − 8·64-s − 12·71-s + 6·73-s − 7·75-s + 81-s − 4·84-s − 4·87-s + 8·89-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.755·7-s + 1/3·9-s − 0.577·12-s + 16-s + 0.436·21-s − 7/5·25-s + 0.192·27-s − 0.755·28-s − 0.742·29-s − 1/3·36-s + 2.43·43-s + 0.577·48-s − 49-s + 3.29·53-s + 0.520·59-s − 0.512·61-s + 0.251·63-s − 64-s − 1.42·71-s + 0.702·73-s − 0.808·75-s + 1/9·81-s − 0.436·84-s − 0.428·87-s + 0.847·89-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

\(L(1)\) \(\approx\) \(1.926392461\)
\(L(\frac12)\) \(\approx\) \(1.926392461\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.5.a_h
7$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ac_l
11$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \) 2.11.a_p
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \) 2.17.a_f
23$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.23.a_o
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.e_ba
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.43.aq_fl
47$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \) 2.47.a_at
53$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.53.ay_jq
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.59.ae_eo
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.61.e_bt
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.67.a_di
71$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.71.m_da
73$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.73.ag_fj
79$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.79.a_abe
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.83.a_dm
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.89.ai_fa
97$C_2^2$ \( 1 + 174 T^{2} + p^{2} T^{4} \) 2.97.a_gs
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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

−8.474357019781163165970864331934, −7.893463375310544986070836175737, −7.64717775810307834107684761356, −7.30383608679715516482480759212, −6.62107654304363600722068227573, −5.90507734663392463514927799475, −5.60328527524921294796493498639, −5.15628820743010086340286347987, −4.49011982263306124172520940094, −4.03703927701628863738367595382, −3.82018858620679634779125359893, −2.99903455583241713754224154253, −2.30027426888824672350698399264, −1.66111190830172932214622271969, −0.71147840329672389227483165322, 0.71147840329672389227483165322, 1.66111190830172932214622271969, 2.30027426888824672350698399264, 2.99903455583241713754224154253, 3.82018858620679634779125359893, 4.03703927701628863738367595382, 4.49011982263306124172520940094, 5.15628820743010086340286347987, 5.60328527524921294796493498639, 5.90507734663392463514927799475, 6.62107654304363600722068227573, 7.30383608679715516482480759212, 7.64717775810307834107684761356, 7.893463375310544986070836175737, 8.474357019781163165970864331934

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