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Properties

Label 2-1792-7.6-c0-0-3
Degree $2$
Conductor $1792$
Sign $1$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 9-s − 2·23-s + 25-s + 49-s + 63-s + 2·71-s − 2·79-s + 81-s − 2·113-s + ⋯
L(s)  = 1  + 7-s + 9-s − 2·23-s + 25-s + 49-s + 63-s + 2·71-s − 2·79-s + 81-s − 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1792} (769, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.347925372\)
\(L(\frac12)\) \(\approx\) \(1.347925372\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( ( 1 + T )^{2} \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.545998237072058705534096772272, −8.556886793704676580779082052564, −7.920802860407620295451947683645, −7.19690687416767209182259427180, −6.31907688205361741916778456464, −5.32502315731445283064871238001, −4.49903972789580943965097259725, −3.79071033929903495789255956714, −2.34758610355770687505163932265, −1.37043159335798875923406158587, 1.37043159335798875923406158587, 2.34758610355770687505163932265, 3.79071033929903495789255956714, 4.49903972789580943965097259725, 5.32502315731445283064871238001, 6.31907688205361741916778456464, 7.19690687416767209182259427180, 7.920802860407620295451947683645, 8.556886793704676580779082052564, 9.545998237072058705534096772272

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