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Properties

Label 2-1792-8.5-c1-0-31
Degree $2$
Conductor $1792$
Sign $0.707 + 0.707i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 3·9-s − 2i·11-s − 4i·13-s − 2·17-s + 4i·19-s + 4·23-s + 5·25-s − 6i·29-s − 8·31-s − 2i·37-s + 2·41-s − 10i·43-s + 49-s + 2i·53-s + ⋯
L(s)  = 1  + 0.377·7-s + 9-s − 0.603i·11-s − 1.10i·13-s − 0.485·17-s + 0.917i·19-s + 0.834·23-s + 25-s − 1.11i·29-s − 1.43·31-s − 0.328i·37-s + 0.312·41-s − 1.52i·43-s + 0.142·49-s + 0.274i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.900251947\)
\(L(\frac12)\) \(\approx\) \(1.900251947\)
\(L(\frac{3}{2})\) 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 \( 1 - 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.135650725998623285290750233005, −8.374125220443382342129593622622, −7.60012811834899552861422360368, −6.92238324424321597406198806993, −5.86926741466675706714715875448, −5.16441386116725114295402616521, −4.15654526657241875889334656083, −3.30223085698716631427980905066, −2.06131068526329584299368979408, −0.805421380845266839850531709462, 1.28389662298130321428503938968, 2.25754894561950131483844248447, 3.55662865214493587313428883551, 4.67946951200927850672233439109, 4.92965654473999670807073524902, 6.44522108096299134878571286584, 7.01832004075914800025445924804, 7.59498820807824465502318407863, 8.878734379490234258949342551132, 9.189786435644848203046447433443

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