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Properties

Label 2-1792-16.13-c1-0-38
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  + (1.23 + 1.23i)3-s + (−0.571 + 0.571i)5-s − i·7-s + 0.0594i·9-s + (−4.55 + 4.55i)11-s + (−3.65 − 3.65i)13-s − 1.41·15-s + 2.47·17-s + (−5.54 − 5.54i)19-s + (1.23 − 1.23i)21-s − 4.08i·23-s + 4.34i·25-s + (3.63 − 3.63i)27-s + (−4.30 − 4.30i)29-s − 1.02·31-s + ⋯
L(s)  = 1  + (0.714 + 0.714i)3-s + (−0.255 + 0.255i)5-s − 0.377i·7-s + 0.0198i·9-s + (−1.37 + 1.37i)11-s + (−1.01 − 1.01i)13-s − 0.365·15-s + 0.599·17-s + (−1.27 − 1.27i)19-s + (0.269 − 0.269i)21-s − 0.851i·23-s + 0.869i·25-s + (0.699 − 0.699i)27-s + (−0.798 − 0.798i)29-s − 0.184·31-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: $\chi_{1792} (1345, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5691757503\)
\(L(\frac12)\) \(\approx\) \(0.5691757503\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 1.23i)T + 3iT^{2} \)
5 \( 1 + (0.571 - 0.571i)T - 5iT^{2} \)
11 \( 1 + (4.55 - 4.55i)T - 11iT^{2} \)
13 \( 1 + (3.65 + 3.65i)T + 13iT^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 + (5.54 + 5.54i)T + 19iT^{2} \)
23 \( 1 + 4.08iT - 23T^{2} \)
29 \( 1 + (4.30 + 4.30i)T + 29iT^{2} \)
31 \( 1 + 1.02T + 31T^{2} \)
37 \( 1 + (-5.91 + 5.91i)T - 37iT^{2} \)
41 \( 1 + 5.47iT - 41T^{2} \)
43 \( 1 + (4.35 - 4.35i)T - 43iT^{2} \)
47 \( 1 - 6.47T + 47T^{2} \)
53 \( 1 + (1.94 - 1.94i)T - 53iT^{2} \)
59 \( 1 + (6.90 - 6.90i)T - 59iT^{2} \)
61 \( 1 + (7.84 + 7.84i)T + 61iT^{2} \)
67 \( 1 + (-3.88 - 3.88i)T + 67iT^{2} \)
71 \( 1 - 9.44iT - 71T^{2} \)
73 \( 1 + 1.06iT - 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (-5.68 - 5.68i)T + 83iT^{2} \)
89 \( 1 + 2.81iT - 89T^{2} \)
97 \( 1 + 1.74T + 97T^{2} \)
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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

−9.232249710952348984505929049651, −8.141989657338919742212405653392, −7.56556078707759016404169067413, −6.93183753638504044637491161279, −5.60937085211253993312239154843, −4.70124485106611759415609498288, −4.11187230494352599696268447910, −2.89057978884609739255938481689, −2.36786759138558752387130254869, −0.17535958429806206975419124703, 1.64086060012169212690409968926, 2.55186737171242122452418280282, 3.40380017246812332173869936671, 4.64256219533322383989569081135, 5.53377434223212171457559543008, 6.35808094583347576373855360131, 7.46191452425557494218732594291, 8.021890957215383936857492052231, 8.469695215820944214304775151047, 9.331120683731496914215625357225

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