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Properties

Label 4-424224-1.1-c1e2-0-2
Degree $4$
Conductor $424224$
Sign $1$
Analytic cond. $27.0488$
Root an. cond. $2.28053$
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 − 8-s + 9-s − 11-s + 12-s − 2·13-s + 16-s − 18-s + 22-s + 6·23-s − 24-s + 2·25-s + 2·26-s + 27-s − 32-s − 33-s + 36-s − 6·37-s − 2·39-s − 44-s − 6·46-s + 11·47-s + 48-s − 4·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.235·18-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 2/5·25-s + 0.392·26-s + 0.192·27-s − 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.986·37-s − 0.320·39-s − 0.150·44-s − 0.884·46-s + 1.60·47-s + 0.144·48-s − 4/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 424224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 424224 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(424224\)    =    \(2^{5} \cdot 3^{3} \cdot 491\)
Sign: $1$
Analytic conductor: \(27.0488\)
Root analytic conductor: \(2.28053\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 424224,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.516664590\)
\(L(\frac12)\) \(\approx\) \(1.516664590\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
491$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 12 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.11.b_c
13$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.13.c_x
17$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.17.a_ag
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.19.a_c
23$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.23.ag_bn
29$C_2^2$ \( 1 + 51 T^{2} + p^{2} T^{4} \) 2.29.a_bz
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.a_bl
37$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.g_t
41$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.41.a_acs
43$C_2^2$ \( 1 - 57 T^{2} + p^{2} T^{4} \) 2.43.a_acf
47$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.al_dq
53$C_2^2$ \( 1 - 69 T^{2} + p^{2} T^{4} \) 2.53.a_acr
59$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.59.ad_dm
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.61.bc_mg
67$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \) 2.67.a_bg
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) 2.71.ao_hi
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) 2.73.ar_ii
79$C_2^2$ \( 1 + 63 T^{2} + p^{2} T^{4} \) 2.79.a_cl
83$C_2$$\times$$C_2$ \( ( 1 - 17 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.aj_be
89$C_2^2$ \( 1 - 117 T^{2} + p^{2} T^{4} \) 2.89.a_aen
97$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.ad_hc
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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

−8.663941634266451530456210002174, −8.199832504768233837369541626515, −7.68874033106993098088605174035, −7.38886488964738417133946143422, −6.94724470371480430003830846554, −6.43168301073555682184370361937, −5.95839289338519467591225258451, −5.11168912458440544968106733640, −4.96990739743716031913330091731, −4.20861392077589236020273548667, −3.44581108192472176022631764464, −3.03299643705175052998264257770, −2.37324745134604451965327576362, −1.73929035573511428656684674832, −0.73431138522739712980354659508, 0.73431138522739712980354659508, 1.73929035573511428656684674832, 2.37324745134604451965327576362, 3.03299643705175052998264257770, 3.44581108192472176022631764464, 4.20861392077589236020273548667, 4.96990739743716031913330091731, 5.11168912458440544968106733640, 5.95839289338519467591225258451, 6.43168301073555682184370361937, 6.94724470371480430003830846554, 7.38886488964738417133946143422, 7.68874033106993098088605174035, 8.199832504768233837369541626515, 8.663941634266451530456210002174

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