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Properties

Label 4-623808-1.1-c1e2-0-13
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 + 9-s − 6·19-s + 6·25-s − 27-s − 6·29-s + 18·41-s − 12·43-s − 10·49-s + 6·53-s + 6·57-s + 8·59-s − 8·61-s + 12·71-s + 8·73-s − 6·75-s + 81-s + 6·87-s − 18·89-s + 4·107-s + 18·113-s + 6·121-s − 18·123-s + 127-s + 12·129-s + 131-s + 137-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.37·19-s + 6/5·25-s − 0.192·27-s − 1.11·29-s + 2.81·41-s − 1.82·43-s − 1.42·49-s + 0.824·53-s + 0.794·57-s + 1.04·59-s − 1.02·61-s + 1.42·71-s + 0.936·73-s − 0.692·75-s + 1/9·81-s + 0.643·87-s − 1.90·89-s + 0.386·107-s + 1.69·113-s + 6/11·121-s − 1.62·123-s + 0.0887·127-s + 1.05·129-s + 0.0873·131-s + 0.0854·137-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

\(L(1)\) \(\approx\) \(1.242050304\)
\(L(\frac12)\) \(\approx\) \(1.242050304\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 6 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.g_co
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.41.as_gg
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.47.a_o
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ag_co
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.i_dy
67$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.67.a_bi
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.am_fm
73$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.ai_ew
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.83.a_co
89$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.s_gw
97$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.97.a_ao
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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

−8.438751586425724428114617365913, −7.955497807542985706446486077888, −7.40915102014633772943899968298, −7.00354104003569786630422972244, −6.52012108211125726403801709815, −6.17460330461658071711152741904, −5.66166980839627965816048490433, −5.13197123038658603513682411362, −4.69005051342033743655650821167, −4.15265363684348228508107750838, −3.68616980551480595374854275616, −2.94662827515129674666916976417, −2.28134437748715567694121300002, −1.60849955023145439407104165383, −0.58907368597405324582529116960, 0.58907368597405324582529116960, 1.60849955023145439407104165383, 2.28134437748715567694121300002, 2.94662827515129674666916976417, 3.68616980551480595374854275616, 4.15265363684348228508107750838, 4.69005051342033743655650821167, 5.13197123038658603513682411362, 5.66166980839627965816048490433, 6.17460330461658071711152741904, 6.52012108211125726403801709815, 7.00354104003569786630422972244, 7.40915102014633772943899968298, 7.955497807542985706446486077888, 8.438751586425724428114617365913

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