Properties

Label 2-1792-16.5-c1-0-26
Degree $2$
Conductor $1792$
Sign $0.991 + 0.130i$
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  + (−1.62 + 1.62i)3-s + (1.16 + 1.16i)5-s + i·7-s − 2.28i·9-s + (−4.39 − 4.39i)11-s + (−0.448 + 0.448i)13-s − 3.80·15-s + 5.02·17-s + (1.49 − 1.49i)19-s + (−1.62 − 1.62i)21-s − 8.89i·23-s − 2.26i·25-s + (−1.15 − 1.15i)27-s + (0.803 − 0.803i)29-s + 8.27·31-s + ⋯
L(s)  = 1  + (−0.938 + 0.938i)3-s + (0.522 + 0.522i)5-s + 0.377i·7-s − 0.762i·9-s + (−1.32 − 1.32i)11-s + (−0.124 + 0.124i)13-s − 0.981·15-s + 1.21·17-s + (0.343 − 0.343i)19-s + (−0.354 − 0.354i)21-s − 1.85i·23-s − 0.453i·25-s + (−0.222 − 0.222i)27-s + (0.149 − 0.149i)29-s + 1.48·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.991 + 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043962590\)
\(L(\frac12)\) \(\approx\) \(1.043962590\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + (1.62 - 1.62i)T - 3iT^{2} \)
5 \( 1 + (-1.16 - 1.16i)T + 5iT^{2} \)
11 \( 1 + (4.39 + 4.39i)T + 11iT^{2} \)
13 \( 1 + (0.448 - 0.448i)T - 13iT^{2} \)
17 \( 1 - 5.02T + 17T^{2} \)
19 \( 1 + (-1.49 + 1.49i)T - 19iT^{2} \)
23 \( 1 + 8.89iT - 23T^{2} \)
29 \( 1 + (-0.803 + 0.803i)T - 29iT^{2} \)
31 \( 1 - 8.27T + 31T^{2} \)
37 \( 1 + (1.55 + 1.55i)T + 37iT^{2} \)
41 \( 1 - 4.93iT - 41T^{2} \)
43 \( 1 + (-3.73 - 3.73i)T + 43iT^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 + (4.53 + 4.53i)T + 53iT^{2} \)
59 \( 1 + (1.06 + 1.06i)T + 59iT^{2} \)
61 \( 1 + (-5.11 + 5.11i)T - 61iT^{2} \)
67 \( 1 + (7.47 - 7.47i)T - 67iT^{2} \)
71 \( 1 + 4.07iT - 71T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 - 9.19T + 79T^{2} \)
83 \( 1 + (2.62 - 2.62i)T - 83iT^{2} \)
89 \( 1 - 1.60iT - 89T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−9.577112148557066026778644851293, −8.420287632074428775244952976842, −7.87431074366464492662380740472, −6.42590024947889799263639663791, −6.03577872568921227026299727870, −5.18536360542830241963942484771, −4.61100154396277036053632078954, −3.21873847647651066496821100202, −2.53149353013510621464677743495, −0.51886778232323286912782672806, 1.06510934044425352687061106727, 1.89387769585994309417674512168, 3.26449917419211464994093638698, 4.70847632622750172053145650781, 5.43868575324245795672365939458, 5.87184607038028890904939811320, 7.17248943982821051589909759948, 7.41439176225283383912249453428, 8.247455001513904328850729592693, 9.640249342072341784801644731399

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