Properties

Label 4-28e2-1.1-c2e2-0-0
Degree $4$
Conductor $784$
Sign $1$
Analytic cond. $0.582084$
Root an. cond. $0.873467$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·7-s − 6·9-s − 12·11-s − 60·23-s + 26·25-s − 12·29-s + 20·37-s + 20·43-s + 51·49-s + 180·53-s − 60·63-s − 140·67-s + 84·71-s − 120·77-s + 148·79-s − 45·81-s + 72·99-s − 300·107-s − 172·109-s + 180·113-s − 134·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯
L(s)  = 1  + 10/7·7-s − 2/3·9-s − 1.09·11-s − 2.60·23-s + 1.03·25-s − 0.413·29-s + 0.540·37-s + 0.465·43-s + 1.04·49-s + 3.39·53-s − 0.952·63-s − 2.08·67-s + 1.18·71-s − 1.55·77-s + 1.87·79-s − 5/9·81-s + 8/11·99-s − 2.80·107-s − 1.57·109-s + 1.59·113-s − 1.10·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.582084\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 784,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9385735968\)
\(L(\frac12)\) \(\approx\) \(0.9385735968\)
\(L(2)\) 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)$
bad2 \( 1 \)
7$C_2$ \( 1 - 10 T + p^{2} T^{2} \)
good3$C_2^2$ \( 1 + 2 p T^{2} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 26 T^{2} + p^{4} T^{4} \)
11$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 314 T^{2} + p^{4} T^{4} \)
17$C_2^2$ \( 1 - 194 T^{2} + p^{4} T^{4} \)
19$C_2^2$ \( 1 - 122 T^{2} + p^{4} T^{4} \)
23$C_2$ \( ( 1 + 30 T + p^{2} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p^{2} T^{2} )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
37$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 962 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 10 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4034 T^{2} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - 90 T + p^{2} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 6362 T^{2} + p^{4} T^{4} \)
61$C_2^2$ \( 1 - 6842 T^{2} + p^{4} T^{4} \)
67$C_2$ \( ( 1 + 70 T + p^{2} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 42 T + p^{2} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 958 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 - 74 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 9722 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 5758 T^{2} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 12674 T^{2} + p^{4} T^{4} \)
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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

−17.49546827430041657887027019916, −16.55687662739985045994637523519, −16.32510798008001579172429449035, −15.36279602132048395755420746153, −14.88408523470219201138873081677, −14.28217893432124476374902789828, −13.74088378107400828707586943021, −13.10212930056448555387090013297, −11.97016493292561868322426830582, −11.88210003694274100340271822057, −10.76355883241477193299422174829, −10.54152099235473083826173450415, −9.482227615847877260948823674436, −8.414074673860621653697820224068, −8.102851334220104601956306921892, −7.28194112582239693475608969263, −5.92187780477507538380273482265, −5.23974565656898765763989074898, −4.14483405066882728762060739027, −2.35788931264915256215796178855, 2.35788931264915256215796178855, 4.14483405066882728762060739027, 5.23974565656898765763989074898, 5.92187780477507538380273482265, 7.28194112582239693475608969263, 8.102851334220104601956306921892, 8.414074673860621653697820224068, 9.482227615847877260948823674436, 10.54152099235473083826173450415, 10.76355883241477193299422174829, 11.88210003694274100340271822057, 11.97016493292561868322426830582, 13.10212930056448555387090013297, 13.74088378107400828707586943021, 14.28217893432124476374902789828, 14.88408523470219201138873081677, 15.36279602132048395755420746153, 16.32510798008001579172429449035, 16.55687662739985045994637523519, 17.49546827430041657887027019916

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