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Properties

Label 2-1792-16.5-c1-0-34
Degree $2$
Conductor $1792$
Sign $0.382 + 0.923i$
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  + (0.898 − 0.898i)3-s + (−0.786 − 0.786i)5-s − i·7-s + 1.38i·9-s + (2.15 + 2.15i)11-s + (1.31 − 1.31i)13-s − 1.41·15-s − 1.79·17-s + (0.531 − 0.531i)19-s + (−0.898 − 0.898i)21-s − 5.95i·23-s − 3.76i·25-s + (3.94 + 3.94i)27-s + (5.62 − 5.62i)29-s + 4.34·31-s + ⋯
L(s)  = 1  + (0.519 − 0.519i)3-s + (−0.351 − 0.351i)5-s − 0.377i·7-s + 0.461i·9-s + (0.650 + 0.650i)11-s + (0.364 − 0.364i)13-s − 0.365·15-s − 0.436·17-s + (0.121 − 0.121i)19-s + (−0.196 − 0.196i)21-s − 1.24i·23-s − 0.752i·25-s + (0.758 + 0.758i)27-s + (1.04 − 1.04i)29-s + 0.779·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.382 + 0.923i$
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: \((2,\ 1792,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.996607311\)
\(L(\frac12)\) \(\approx\) \(1.996607311\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.898 + 0.898i)T - 3iT^{2} \)
5 \( 1 + (0.786 + 0.786i)T + 5iT^{2} \)
11 \( 1 + (-2.15 - 2.15i)T + 11iT^{2} \)
13 \( 1 + (-1.31 + 1.31i)T - 13iT^{2} \)
17 \( 1 + 1.79T + 17T^{2} \)
19 \( 1 + (-0.531 + 0.531i)T - 19iT^{2} \)
23 \( 1 + 5.95iT - 23T^{2} \)
29 \( 1 + (-5.62 + 5.62i)T - 29iT^{2} \)
31 \( 1 - 4.34T + 31T^{2} \)
37 \( 1 + (-2.12 - 2.12i)T + 37iT^{2} \)
41 \( 1 + 0.712iT - 41T^{2} \)
43 \( 1 + (-2.96 - 2.96i)T + 43iT^{2} \)
47 \( 1 + 2.20T + 47T^{2} \)
53 \( 1 + (3.38 + 3.38i)T + 53iT^{2} \)
59 \( 1 + (-2.41 - 2.41i)T + 59iT^{2} \)
61 \( 1 + (-7.09 + 7.09i)T - 61iT^{2} \)
67 \( 1 + (6.76 - 6.76i)T - 67iT^{2} \)
71 \( 1 + 1.96iT - 71T^{2} \)
73 \( 1 + 10.3iT - 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (-5.95 + 5.95i)T - 83iT^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−8.873443955002066960186279391815, −8.294275550576937525967208112153, −7.70971209616009448669492220023, −6.80318720934603725223302958435, −6.16143734732640324568357734653, −4.70186571447467487535471944237, −4.34415478212565227264089248980, −3.00606595206472097531870953627, −2.06247234270350055687480483675, −0.805338175732221456827006526963, 1.25118480229709264252084778326, 2.78798807466425500236249501246, 3.53722538114609763713137311634, 4.20804696553781547600240317975, 5.37303589151494067169627706982, 6.32089114321547542395532201886, 6.98027080918398831799070105093, 8.027163768168531030727228173190, 8.839298942750100364601556774677, 9.230846743663109626426868658473

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