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Properties

Label 4-528e2-1.1-c1e2-0-46
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  − 2·3-s + 9-s + 2·11-s − 8·17-s − 4·25-s + 4·27-s + 12·29-s − 4·33-s + 4·37-s − 4·41-s − 8·49-s + 16·51-s − 16·67-s + 8·75-s − 11·81-s − 24·87-s + 8·97-s + 2·99-s + 12·101-s + 16·103-s + 4·107-s − 8·111-s − 7·121-s + 8·123-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 0.603·11-s − 1.94·17-s − 4/5·25-s + 0.769·27-s + 2.22·29-s − 0.696·33-s + 0.657·37-s − 0.624·41-s − 8/7·49-s + 2.24·51-s − 1.95·67-s + 0.923·75-s − 1.22·81-s − 2.57·87-s + 0.812·97-s + 0.201·99-s + 1.19·101-s + 1.57·103-s + 0.386·107-s − 0.759·111-s − 0.636·121-s + 0.721·123-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$C_2$ \( 1 + 2 T + p T^{2} \)
11$C_2$ \( 1 - 2 T + p T^{2} \)
good5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
7$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.7.a_i
13$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.13.a_e
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.17.i_by
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.19.a_k
23$C_2^2$ \( 1 - 24 T^{2} + p^{2} T^{4} \) 2.23.a_ay
29$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.am_da
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.e_w
43$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.43.a_s
47$C_2^2$ \( 1 - 40 T^{2} + p^{2} T^{4} \) 2.47.a_abo
53$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \) 2.53.a_cq
59$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.59.a_acs
61$C_2^2$ \( 1 - 76 T^{2} + p^{2} T^{4} \) 2.61.a_acy
67$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.67.q_ha
71$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.71.a_i
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.73.a_eg
79$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \) 2.79.a_i
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.89.a_as
97$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.97.ai_fq
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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

−8.735424743969913722401442704102, −8.260358573332115025179939055540, −7.63709910505436727931709519719, −7.07612960861449027506487707789, −6.48798309207190659245826576807, −6.33701902574578098603648188958, −5.95596886629459125001106908776, −5.15237028527185683958489699484, −4.59839678696927045165011026513, −4.49090593511407312384272154710, −3.62078744508708309990334939125, −2.84975063122556541304592322167, −2.13616819216613900678717943152, −1.16145006700948142253607225982, 0, 1.16145006700948142253607225982, 2.13616819216613900678717943152, 2.84975063122556541304592322167, 3.62078744508708309990334939125, 4.49090593511407312384272154710, 4.59839678696927045165011026513, 5.15237028527185683958489699484, 5.95596886629459125001106908776, 6.33701902574578098603648188958, 6.48798309207190659245826576807, 7.07612960861449027506487707789, 7.63709910505436727931709519719, 8.260358573332115025179939055540, 8.735424743969913722401442704102

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