Properties

Label 4-1792-1.1-c1e2-0-0
Degree $4$
Conductor $1792$
Sign $1$
Analytic cond. $0.114259$
Root an. cond. $0.581397$
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  − 3·7-s − 2·9-s + 2·25-s + 4·31-s − 12·41-s + 12·47-s + 6·49-s + 6·63-s − 12·71-s + 16·73-s + 4·79-s − 5·81-s − 8·97-s + 16·103-s − 12·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + ⋯
L(s)  = 1  − 1.13·7-s − 2/3·9-s + 2/5·25-s + 0.718·31-s − 1.87·41-s + 1.75·47-s + 6/7·49-s + 0.755·63-s − 1.42·71-s + 1.87·73-s + 0.450·79-s − 5/9·81-s − 0.812·97-s + 1.57·103-s − 1.12·113-s + 2/11·121-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 + 0.0783·163-s + 0.0773·167-s + 2/13·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.114259\)
Root analytic conductor: \(0.581397\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1792} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1792,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5715476195\)
\(L(\frac12)\) \(\approx\) \(0.5715476195\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 4 T + p T^{2} ) \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
53$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
71$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−13.50168732309307683260627329345, −12.86901379746632539235112958179, −12.19325986489256484230240936608, −11.76031412251701343638631858204, −10.89959748266339602534578702656, −10.28638500753365554891315811069, −9.678099476978539892491368486264, −8.928595102227002440896862141069, −8.387252653197497723949103538111, −7.39515097778537885145910103711, −6.62947344847081355188711857038, −5.96486295797288004275263855106, −5.03435508364857530236755864838, −3.76784593849749485024605323535, −2.75646022749770999248825075408, 2.75646022749770999248825075408, 3.76784593849749485024605323535, 5.03435508364857530236755864838, 5.96486295797288004275263855106, 6.62947344847081355188711857038, 7.39515097778537885145910103711, 8.387252653197497723949103538111, 8.928595102227002440896862141069, 9.678099476978539892491368486264, 10.28638500753365554891315811069, 10.89959748266339602534578702656, 11.76031412251701343638631858204, 12.19325986489256484230240936608, 12.86901379746632539235112958179, 13.50168732309307683260627329345

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