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Properties

Label 4-528e2-1.1-c1e2-0-42
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 $0$

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 − 2·29-s + 6·31-s + 32-s + 4·34-s + 36-s + 12·37-s − 8·41-s + 2·48-s + 6·49-s + 2·50-s + 8·51-s − 4·54-s − 2·58-s + 6·62-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 − 0.371·29-s + 1.07·31-s + 0.176·32-s + 0.685·34-s + 1/6·36-s + 1.97·37-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 − 0.262·58-s + 0.762·62-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: \(0\)
Selberg data: \((4,\ 278784,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.347110763\)
\(L(\frac12)\) \(\approx\) \(4.347110763\)
\(L(\frac{3}{2})\) 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_ac
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.13.a_as
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_aba
29$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.c_by
31$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.31.ag_ck
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.i_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_ck
53$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.53.a_aby
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_dy
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_aco
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_acc
83$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.83.ai_eo
89$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.89.a_acs
97$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.e_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

−8.865739497167248133100294373314, −8.269850318251430758878581823136, −7.910856626788442893756335600532, −7.67149384481827260878738860184, −6.93079453766029115486024561314, −6.59859930725279222005763808806, −5.82033637819644646770339581371, −5.60340714340328404348864825659, −4.82123575336124461511639695635, −4.34830377773187484391964833484, −3.71414385623981426766906186704, −3.21853192430282872984834774076, −2.70267444048733703533689861898, −2.11956303338550843699547167488, −1.12907565963860211137938267625, 1.12907565963860211137938267625, 2.11956303338550843699547167488, 2.70267444048733703533689861898, 3.21853192430282872984834774076, 3.71414385623981426766906186704, 4.34830377773187484391964833484, 4.82123575336124461511639695635, 5.60340714340328404348864825659, 5.82033637819644646770339581371, 6.59859930725279222005763808806, 6.93079453766029115486024561314, 7.67149384481827260878738860184, 7.910856626788442893756335600532, 8.269850318251430758878581823136, 8.865739497167248133100294373314

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