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Properties

Label 4-528e2-1.1-c1e2-0-27
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  + 3-s + 3·7-s − 2·9-s + 5·11-s − 7·13-s + 3·21-s + 8·25-s − 5·27-s + 5·29-s + 5·33-s − 7·39-s − 49-s + 10·59-s + 4·61-s − 6·63-s + 15·67-s + 8·75-s + 15·77-s − 11·79-s + 81-s + 5·87-s + 5·89-s − 21·91-s + 8·97-s − 10·99-s + 5·101-s + 32·109-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.50·11-s − 1.94·13-s + 0.654·21-s + 8/5·25-s − 0.962·27-s + 0.928·29-s + 0.870·33-s − 1.12·39-s − 1/7·49-s + 1.30·59-s + 0.512·61-s − 0.755·63-s + 1.83·67-s + 0.923·75-s + 1.70·77-s − 1.23·79-s + 1/9·81-s + 0.536·87-s + 0.529·89-s − 2.20·91-s + 0.812·97-s − 1.00·99-s + 0.497·101-s + 3.06·109-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

\(L(1)\) \(\approx\) \(2.404316561\)
\(L(\frac12)\) \(\approx\) \(2.404316561\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - T + p T^{2} \)
11$C_2$ \( 1 - 5 T + p T^{2} \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) 2.7.ad_k
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.h_bg
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 + 33 T^{2} + p^{2} T^{4} \) 2.23.a_bh
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.29.af_i
31$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.31.a_u
37$C_2^2$ \( 1 + 51 T^{2} + p^{2} T^{4} \) 2.37.a_bz
41$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \) 2.41.a_aq
43$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \) 2.43.a_cm
47$C_2^2$ \( 1 - 9 T^{2} + p^{2} T^{4} \) 2.47.a_aj
53$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.53.a_bc
59$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.59.ak_fm
61$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.61.ae_dx
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) 2.67.ap_hc
71$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.71.a_aby
73$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.73.a_h
79$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.79.l_fc
83$C_2^2$ \( 1 + 39 T^{2} + p^{2} T^{4} \) 2.83.a_bn
89$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.af_gw
97$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.97.ai_ic
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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

−8.848782040696916311947200734605, −8.313926750711264222217888453128, −8.194237735785393763239551957462, −7.42840894357210660099632843527, −7.06096615598236951516220087416, −6.67123063506885095674961719113, −6.02377202930060645154318785253, −5.33560763689611435449542913537, −4.85026919917921393990174230208, −4.56890516246816681335624547661, −3.80812694740761668030501043183, −3.15829399956755989167188739966, −2.48465184837308920909785594054, −1.95834330304523209911742405938, −0.930958291469088077999067927044, 0.930958291469088077999067927044, 1.95834330304523209911742405938, 2.48465184837308920909785594054, 3.15829399956755989167188739966, 3.80812694740761668030501043183, 4.56890516246816681335624547661, 4.85026919917921393990174230208, 5.33560763689611435449542913537, 6.02377202930060645154318785253, 6.67123063506885095674961719113, 7.06096615598236951516220087416, 7.42840894357210660099632843527, 8.194237735785393763239551957462, 8.313926750711264222217888453128, 8.848782040696916311947200734605

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