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Properties

Label 4-896e2-1.1-c1e2-0-9
Degree $4$
Conductor $802816$
Sign $1$
Analytic cond. $51.1882$
Root an. cond. $2.67480$
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  + 4·7-s − 4·9-s + 4·11-s − 4·25-s + 6·29-s + 10·37-s − 12·43-s + 9·49-s + 10·53-s − 16·63-s + 12·67-s + 8·71-s + 16·77-s − 16·79-s + 7·81-s − 16·99-s − 4·107-s + 18·109-s + 8·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
L(s)  = 1  + 1.51·7-s − 4/3·9-s + 1.20·11-s − 4/5·25-s + 1.11·29-s + 1.64·37-s − 1.82·43-s + 9/7·49-s + 1.37·53-s − 2.01·63-s + 1.46·67-s + 0.949·71-s + 1.82·77-s − 1.80·79-s + 7/9·81-s − 1.60·99-s − 0.386·107-s + 1.72·109-s + 0.752·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-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: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 802816,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380631337\)
\(L(\frac12)\) \(\approx\) \(2.380631337\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.5.a_e
11$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.11.ae_w
13$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.13.a_u
17$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.17.a_ao
19$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.19.a_e
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.23.a_be
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.29.ag_co
31$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.31.a_s
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.37.ak_du
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.47.a_ao
53$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.53.ak_fa
59$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.59.a_u
61$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.61.a_ado
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.67.am_gk
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.ai_dq
73$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.73.a_abi
79$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.q_hy
83$C_2^2$ \( 1 + 52 T^{2} + p^{2} T^{4} \) 2.83.a_ca
89$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.89.a_ac
97$C_2^2$ \( 1 + 146 T^{2} + p^{2} T^{4} \) 2.97.a_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

−8.240933765556874802038078268283, −7.978791796060660420870812430238, −7.44608252878462617373329558634, −6.85837606570585725941475805396, −6.46252920356004503364615799287, −5.93129613082725659790104036681, −5.56087131145895171569226375181, −5.02417808289938285535275447904, −4.57934355049213146946943272399, −4.07013116279426977050235252680, −3.54433570013807539633403039543, −2.83031760233214085553452092368, −2.24208405724219605019537317045, −1.58515595650476848472698733656, −0.78210697286726853691608896594, 0.78210697286726853691608896594, 1.58515595650476848472698733656, 2.24208405724219605019537317045, 2.83031760233214085553452092368, 3.54433570013807539633403039543, 4.07013116279426977050235252680, 4.57934355049213146946943272399, 5.02417808289938285535275447904, 5.56087131145895171569226375181, 5.93129613082725659790104036681, 6.46252920356004503364615799287, 6.85837606570585725941475805396, 7.44608252878462617373329558634, 7.978791796060660420870812430238, 8.240933765556874802038078268283

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