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Properties

Label 4-896e2-1.1-c1e2-0-15
Degree $4$
Conductor $802816$
Sign $-1$
Analytic cond. $51.1882$
Root an. cond. $2.67480$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 4·7-s − 2·9-s − 4·11-s − 4·13-s + 2·25-s − 16·35-s + 12·43-s + 8·45-s + 16·47-s + 9·49-s + 16·55-s − 4·61-s − 8·63-s + 16·65-s + 20·67-s − 16·77-s − 5·81-s − 16·91-s + 8·99-s + 12·101-s + 8·103-s − 4·107-s + 4·113-s + 8·117-s − 10·121-s + 28·125-s + ⋯
L(s)  = 1  − 1.78·5-s + 1.51·7-s − 2/3·9-s − 1.20·11-s − 1.10·13-s + 2/5·25-s − 2.70·35-s + 1.82·43-s + 1.19·45-s + 2.33·47-s + 9/7·49-s + 2.15·55-s − 0.512·61-s − 1.00·63-s + 1.98·65-s + 2.44·67-s − 1.82·77-s − 5/9·81-s − 1.67·91-s + 0.804·99-s + 1.19·101-s + 0.788·103-s − 0.386·107-s + 0.376·113-s + 0.739·117-s − 0.909·121-s + 2.50·125-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 802816 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(802816\)    =    \(2^{14} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(51.1882\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 802816,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.3.a_c
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.5.e_o
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.19.a_bi
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.a_aba
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.43.am_es
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.47.aq_gc
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.59.a_ada
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.67.au_ja
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.a_ac
73$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.73.a_aby
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.a_dq
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.83.a_fa
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.89.a_gs
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−8.098681382024412731658064042949, −7.60669620667117230334580612361, −7.33738117792255002879146989083, −7.08458527518240061849455651795, −6.05418523802166600744858353834, −5.67939354494067973280289925981, −5.16542238567969836254694300670, −4.77922937487002835281324915984, −4.27419960188694524212471687413, −3.91058908136761919842529082491, −3.25684639917526338714188955496, −2.35648801922349828239873492037, −2.30676446591740616355718333113, −0.911211315540143404857619909068, 0, 0.911211315540143404857619909068, 2.30676446591740616355718333113, 2.35648801922349828239873492037, 3.25684639917526338714188955496, 3.91058908136761919842529082491, 4.27419960188694524212471687413, 4.77922937487002835281324915984, 5.16542238567969836254694300670, 5.67939354494067973280289925981, 6.05418523802166600744858353834, 7.08458527518240061849455651795, 7.33738117792255002879146989083, 7.60669620667117230334580612361, 8.098681382024412731658064042949

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