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Properties

Label 4-623808-1.1-c1e2-0-38
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  + 2·2-s + 3-s + 2·4-s + 2·6-s + 2·7-s + 9-s + 2·12-s + 4·14-s − 4·16-s + 2·18-s + 4·19-s + 2·21-s + 3·25-s + 27-s + 4·28-s − 2·29-s − 8·32-s + 2·36-s + 8·38-s + 4·41-s + 4·42-s − 4·48-s − 11·49-s + 6·50-s + 16·53-s + 2·54-s + 4·57-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 0.755·7-s + 1/3·9-s + 0.577·12-s + 1.06·14-s − 16-s + 0.471·18-s + 0.917·19-s + 0.436·21-s + 3/5·25-s + 0.192·27-s + 0.755·28-s − 0.371·29-s − 1.41·32-s + 1/3·36-s + 1.29·38-s + 0.624·41-s + 0.617·42-s − 0.577·48-s − 1.57·49-s + 0.848·50-s + 2.19·53-s + 0.272·54-s + 0.529·57-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

\(L(1)\) \(\approx\) \(5.824134090\)
\(L(\frac12)\) \(\approx\) \(5.824134090\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
3$C_1$ \( 1 - T \)
19$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2^2$ \( 1 + 9 T^{2} + p^{2} T^{4} \) 2.11.a_j
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.13.a_ac
17$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.17.a_av
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.29.c_cg
31$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.31.a_bi
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.37.a_w
41$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.ae_w
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.43.a_cj
47$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \) 2.47.a_cl
53$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.aq_fy
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ac_dq
61$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.61.ae_ev
67$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \) 2.67.a_de
71$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.71.ak_gk
73$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.73.as_hj
79$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.79.a_ade
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.ae_gw
97$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.97.a_abu
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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

−8.299466427781751772702058069200, −7.902426289993434067499557498473, −7.43580314438238289009468748587, −6.92260293458617276722934127091, −6.57415097738785205536396464637, −5.97649668236063640497314631594, −5.41179770483773404560678581344, −5.08450568548600867841604912196, −4.70508723749341815073226066792, −4.04044875364450495814050379668, −3.66007720102416136107079042171, −3.14837060353887846404013102044, −2.47184089781820374192931322175, −2.00160790434408989931509710178, −0.988971569178842219271303108319, 0.988971569178842219271303108319, 2.00160790434408989931509710178, 2.47184089781820374192931322175, 3.14837060353887846404013102044, 3.66007720102416136107079042171, 4.04044875364450495814050379668, 4.70508723749341815073226066792, 5.08450568548600867841604912196, 5.41179770483773404560678581344, 5.97649668236063640497314631594, 6.57415097738785205536396464637, 6.92260293458617276722934127091, 7.43580314438238289009468748587, 7.902426289993434067499557498473, 8.299466427781751772702058069200

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