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Properties

Label 4-528e2-1.1-c1e2-0-45
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  − 4·5-s + 9-s + 2·25-s + 4·37-s − 4·45-s + 6·49-s + 12·53-s + 81-s − 28·89-s − 4·97-s − 12·113-s − 11·121-s + 28·125-s + ⋯
L(s)  = 1  − 1.78·5-s + 1/3·9-s + 2/5·25-s + 0.657·37-s − 0.596·45-s + 6/7·49-s + 1.64·53-s + 1/9·81-s − 2.96·89-s − 0.406·97-s − 1.12·113-s − 121-s + 2.50·125-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: \(1\)
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_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_2$ \( 1 + p T^{2} \)
good5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.19.a_s
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.29.a_abm
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.41.a_ack
43$C_2^2$ \( 1 + 66 T^{2} + p^{2} T^{4} \) 2.43.a_co
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.53.am_fm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.61.a_abq
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.a_ak
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.73.a_aco
79$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.79.a_aw
83$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.83.a_di
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.89.bc_ok
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.97.e_hq
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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

−8.473773067721677742175756371560, −8.182016635415132847970422388821, −7.63226343073355306933742502764, −7.40196904121467042757665636409, −6.90211781992944844765322670884, −6.36561807406552790070698984228, −5.66446886960789635572122358861, −5.24118763659557674114177794867, −4.42225581141460259753947239475, −4.06769646235795927004687967337, −3.77418719387206308424761626324, −2.99232769786698318240568176297, −2.29024981342957591674944719903, −1.15794953445657171983139776026, 0, 1.15794953445657171983139776026, 2.29024981342957591674944719903, 2.99232769786698318240568176297, 3.77418719387206308424761626324, 4.06769646235795927004687967337, 4.42225581141460259753947239475, 5.24118763659557674114177794867, 5.66446886960789635572122358861, 6.36561807406552790070698984228, 6.90211781992944844765322670884, 7.40196904121467042757665636409, 7.63226343073355306933742502764, 8.182016635415132847970422388821, 8.473773067721677742175756371560

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