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Properties

Label 2-1792-8.5-c1-0-39
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  − 2i·3-s + 7-s − 9-s + 4i·11-s − 4i·13-s − 2·17-s − 6i·19-s − 2i·21-s − 8·23-s + 5·25-s − 4i·27-s + 2i·29-s − 4·31-s + 8·33-s − 10i·37-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.377·7-s − 0.333·9-s + 1.20i·11-s − 1.10i·13-s − 0.485·17-s − 1.37i·19-s − 0.436i·21-s − 1.66·23-s + 25-s − 0.769i·27-s + 0.371i·29-s − 0.718·31-s + 1.39·33-s − 1.64i·37-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

\(L(1)\) \(\approx\) \(1.452839816\)
\(L(\frac12)\) \(\approx\) \(1.452839816\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 10iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 4T + 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.886822846202992888782878355008, −7.946775133717328822803951042244, −7.39154135809763517554922142448, −6.82433041023246349280288512703, −5.87072247714935711127050846045, −4.94178959045265490855925218119, −4.02943360239990204590868483715, −2.56579511387070637701763941501, −1.88291078226632566944676563473, −0.54339113468104978389815767993, 1.50414249428293699898402646599, 2.89069247004404218362813725589, 4.06853526969133775380481934010, 4.30876348263583370054341056486, 5.54633344390425667840264944717, 6.14704461923121500029274219723, 7.23902651652528035790225865926, 8.291998783204522234294349930134, 8.785723803933609650617371643745, 9.672732593597194950984025557283

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