Properties

Label 2-1792-8.3-c2-0-92
Degree $2$
Conductor $1792$
Sign $-0.707 - 0.707i$
Analytic cond. $48.8284$
Root an. cond. $6.98773$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.554·3-s − 4.57i·5-s − 2.64i·7-s − 8.69·9-s − 15.7·11-s − 8.57i·13-s + 2.53i·15-s + 28.3·17-s + 6.33·19-s + 1.46i·21-s − 31.0i·23-s + 4.05·25-s + 9.81·27-s + 0.846i·29-s − 21.6i·31-s + ⋯
L(s)  = 1  − 0.184·3-s − 0.915i·5-s − 0.377i·7-s − 0.965·9-s − 1.43·11-s − 0.659i·13-s + 0.169i·15-s + 1.66·17-s + 0.333·19-s + 0.0698i·21-s − 1.35i·23-s + 0.162·25-s + 0.363·27-s + 0.0291i·29-s − 0.697i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(48.8284\)
Root analytic conductor: \(6.98773\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2830899696\)
\(L(\frac12)\) \(\approx\) \(0.2830899696\)
\(L(2)\) 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 \( 1 + 2.64iT \)
good3 \( 1 + 0.554T + 9T^{2} \)
5 \( 1 + 4.57iT - 25T^{2} \)
11 \( 1 + 15.7T + 121T^{2} \)
13 \( 1 + 8.57iT - 169T^{2} \)
17 \( 1 - 28.3T + 289T^{2} \)
19 \( 1 - 6.33T + 361T^{2} \)
23 \( 1 + 31.0iT - 529T^{2} \)
29 \( 1 - 0.846iT - 841T^{2} \)
31 \( 1 + 21.6iT - 961T^{2} \)
37 \( 1 + 33.6iT - 1.36e3T^{2} \)
41 \( 1 + 66.9T + 1.68e3T^{2} \)
43 \( 1 + 44.8T + 1.84e3T^{2} \)
47 \( 1 - 38.4iT - 2.20e3T^{2} \)
53 \( 1 + 14.8iT - 2.80e3T^{2} \)
59 \( 1 + 5.80T + 3.48e3T^{2} \)
61 \( 1 - 52.6iT - 3.72e3T^{2} \)
67 \( 1 + 117.T + 4.48e3T^{2} \)
71 \( 1 - 81.2iT - 5.04e3T^{2} \)
73 \( 1 - 47.8T + 5.32e3T^{2} \)
79 \( 1 - 57.4iT - 6.24e3T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 - 89.2T + 7.92e3T^{2} \)
97 \( 1 + 3.44T + 9.40e3T^{2} \)
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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

−8.269716589839348014620314999159, −8.155081197557696410101506960361, −7.14490039019688954740033184684, −5.88648589129951081459675987336, −5.35302160834667563224909038418, −4.71748606665839718645234246237, −3.39730028786017277342229358647, −2.58263382501433454249003172484, −1.02078915903581091519949678320, −0.085346640442176882852302135767, 1.70863459729737546247399340908, 3.07620286292336150794311298990, 3.23583500997218240039798817529, 5.02318460513406837584886160891, 5.47282305016028842563634155897, 6.36403278393691310286355870112, 7.26327705440997422269593898598, 7.965486133780127228073740811324, 8.700161679271210726149554688903, 9.772177795743176088538410477091

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