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Properties

Label 4-528e2-1.1-c1e2-0-41
Degree $4$
Conductor $278784$
Sign $-1$
Analytic cond. $17.7755$
Root an. cond. $2.05331$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 2·12-s + 16-s + 4·17-s − 18-s + 2·24-s − 2·25-s + 4·27-s − 6·29-s − 6·31-s − 32-s − 4·34-s + 36-s + 8·41-s − 2·48-s − 6·49-s + 2·50-s − 8·51-s − 4·54-s + 6·58-s + 6·62-s + 64-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.408·24-s − 2/5·25-s + 0.769·27-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.24·41-s − 0.288·48-s − 6/7·49-s + 0.282·50-s − 1.12·51-s − 0.544·54-s + 0.787·58-s + 0.762·62-s + 1/8·64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 278784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(278784\)    =    \(2^{8} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(17.7755\)
Root analytic conductor: \(2.05331\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.17.ae_bi
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.19.a_abi
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
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$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.31.g_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.a_bm
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.41.ai_dq
43$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.43.a_ag
47$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.47.a_ack
53$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.53.a_by
59$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.59.a_bq
61$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.61.a_ady
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.a_cs
71$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.71.a_co
73$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.73.a_dm
79$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \) 2.79.a_cc
83$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.i_eo
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.89.a_acs
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.97.ae_fe
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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

−8.694573873478892042367830887308, −8.118713976519692338934094597501, −7.65734009264984512871750918202, −7.29427550763121838826276530791, −6.78425039361168647489035597064, −6.14720207918552725310768300204, −5.84318555834654134787269596842, −5.39424192407206185701036251630, −4.90819948415861353150359458313, −4.12477070460604156991586964316, −3.53389448567225503028918420246, −2.82263018073438078907993017399, −1.94586579326756091315637388338, −1.10515668947707679547612438627, 0, 1.10515668947707679547612438627, 1.94586579326756091315637388338, 2.82263018073438078907993017399, 3.53389448567225503028918420246, 4.12477070460604156991586964316, 4.90819948415861353150359458313, 5.39424192407206185701036251630, 5.84318555834654134787269596842, 6.14720207918552725310768300204, 6.78425039361168647489035597064, 7.29427550763121838826276530791, 7.65734009264984512871750918202, 8.118713976519692338934094597501, 8.694573873478892042367830887308

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