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Properties

Label 4-528e2-1.1-c1e2-0-33
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  + 2·5-s + 2·7-s − 9-s + 4·11-s − 4·19-s + 2·25-s + 4·35-s + 8·37-s − 4·43-s − 2·45-s − 2·49-s + 6·53-s + 8·55-s − 2·63-s + 8·77-s + 10·79-s + 81-s + 8·83-s + 12·89-s − 8·95-s + 28·97-s − 4·99-s − 20·107-s + 12·113-s + 5·121-s + 10·125-s + ⋯
L(s)  = 1  + 0.894·5-s + 0.755·7-s − 1/3·9-s + 1.20·11-s − 0.917·19-s + 2/5·25-s + 0.676·35-s + 1.31·37-s − 0.609·43-s − 0.298·45-s − 2/7·49-s + 0.824·53-s + 1.07·55-s − 0.251·63-s + 0.911·77-s + 1.12·79-s + 1/9·81-s + 0.878·83-s + 1.27·89-s − 0.820·95-s + 2.84·97-s − 0.402·99-s − 1.93·107-s + 1.12·113-s + 5/11·121-s + 0.894·125-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

\(L(1)\) \(\approx\) \(2.445004566\)
\(L(\frac12)\) \(\approx\) \(2.445004566\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.ac_c
7$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.ac_g
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.13.a_o
17$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.17.a_g
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.31.a_abi
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.ai_cc
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.41.a_s
43$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.e_cc
47$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.47.a_aco
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) 2.53.ag_ec
59$C_2^2$ \( 1 + 98 T^{2} + p^{2} T^{4} \) 2.59.a_du
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.61.a_aby
67$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.67.a_aba
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.ak_dy
83$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.83.ai_bm
89$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.89.am_gw
97$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.97.abc_pa
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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

−8.943811935412166881493279763655, −8.529673587212088883260523678465, −7.87509654823026938444447155331, −7.65519451864226378019068035093, −6.76383606529349135945033583766, −6.48863492843520063543136396597, −6.10020868733469769232995596080, −5.51297994615955272595668101965, −4.97138925993982967920085300834, −4.45965116583721194402214423366, −3.89563431077778991339671102073, −3.22778806516645631510592926454, −2.32542314565977026199159319157, −1.88676730418402426868026242938, −0.994416628256912978162949866361, 0.994416628256912978162949866361, 1.88676730418402426868026242938, 2.32542314565977026199159319157, 3.22778806516645631510592926454, 3.89563431077778991339671102073, 4.45965116583721194402214423366, 4.97138925993982967920085300834, 5.51297994615955272595668101965, 6.10020868733469769232995596080, 6.48863492843520063543136396597, 6.76383606529349135945033583766, 7.65519451864226378019068035093, 7.87509654823026938444447155331, 8.529673587212088883260523678465, 8.943811935412166881493279763655

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