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Properties

Label 4-528e2-1.1-c1e2-0-51
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  − 3-s − 2·9-s − 2·11-s − 2·17-s + 3·25-s + 5·27-s + 8·29-s − 6·31-s + 2·33-s − 6·37-s + 6·41-s + 2·49-s + 2·51-s − 2·67-s − 3·75-s + 81-s + 4·83-s − 8·87-s + 6·93-s − 6·97-s + 4·99-s − 2·101-s − 16·103-s − 14·107-s + 6·111-s − 7·121-s − 6·123-s + ⋯
L(s)  = 1  − 0.577·3-s − 2/3·9-s − 0.603·11-s − 0.485·17-s + 3/5·25-s + 0.962·27-s + 1.48·29-s − 1.07·31-s + 0.348·33-s − 0.986·37-s + 0.937·41-s + 2/7·49-s + 0.280·51-s − 0.244·67-s − 0.346·75-s + 1/9·81-s + 0.439·83-s − 0.857·87-s + 0.622·93-s − 0.609·97-s + 0.402·99-s − 0.199·101-s − 1.57·103-s − 1.35·107-s + 0.569·111-s − 0.636·121-s − 0.541·123-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 \( 1 \)
3$C_2$ \( 1 + T + p T^{2} \)
11$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.5.a_ad
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.c_bi
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.23.a_abf
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ai_cs
31$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.31.g_cp
37$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.37.g_cp
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.41.ag_co
43$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.43.a_aco
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.53.a_bi
59$C_2^2$ \( 1 - 55 T^{2} + p^{2} T^{4} \) 2.59.a_acd
61$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.61.a_k
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.67.c_dv
71$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \) 2.71.a_cz
73$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.73.a_ack
79$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.79.a_ao
83$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.ae_ec
89$C_2^2$ \( 1 + 133 T^{2} + p^{2} T^{4} \) 2.89.a_fd
97$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.g_hf
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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

−8.546024841400564250753782968559, −8.281160825029880156167531320771, −7.73962365214990528759464055979, −7.12492841896964221416453524341, −6.72133789258784628138813540346, −6.27140324872629859210074300607, −5.66495576310433969778274580872, −5.29145801525587722605641808960, −4.81133899028676887021276795332, −4.23364921059678854690289116635, −3.50226421438709116849625403522, −2.80869064149029541190941900706, −2.33204853817642631683494677063, −1.19827236084558462850424593227, 0, 1.19827236084558462850424593227, 2.33204853817642631683494677063, 2.80869064149029541190941900706, 3.50226421438709116849625403522, 4.23364921059678854690289116635, 4.81133899028676887021276795332, 5.29145801525587722605641808960, 5.66495576310433969778274580872, 6.27140324872629859210074300607, 6.72133789258784628138813540346, 7.12492841896964221416453524341, 7.73962365214990528759464055979, 8.281160825029880156167531320771, 8.546024841400564250753782968559

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