Properties

Label 2-1792-28.27-c1-0-46
Degree $2$
Conductor $1792$
Sign $-0.755 - 0.654i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.73·3-s − 0.732i·5-s + (−2 − 1.73i)7-s + 4.46·9-s + 5.46i·11-s − 4.73i·13-s + 2i·15-s − 4i·17-s − 1.26·19-s + (5.46 + 4.73i)21-s − 5.46i·23-s + 4.46·25-s − 3.99·27-s − 6.92·29-s + 6.92·31-s + ⋯
L(s)  = 1  − 1.57·3-s − 0.327i·5-s + (−0.755 − 0.654i)7-s + 1.48·9-s + 1.64i·11-s − 1.31i·13-s + 0.516i·15-s − 0.970i·17-s − 0.290·19-s + (1.19 + 1.03i)21-s − 1.13i·23-s + 0.892·25-s − 0.769·27-s − 1.28·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.755 - 0.654i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ -0.755 - 0.654i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + 1.73i)T \)
good3 \( 1 + 2.73T + 3T^{2} \)
5 \( 1 + 0.732iT - 5T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + 4.73iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 2.92iT - 41T^{2} \)
43 \( 1 + 2.53iT - 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + 3.80T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 2.53iT - 67T^{2} \)
71 \( 1 + 4.53iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + 10.7T + 83T^{2} \)
89 \( 1 - 14.9iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−8.956737695596008812412110978882, −7.71246610596543518542251050247, −6.98322139160390847408938641272, −6.48725135068307125875538137542, −5.42232707337653219633245498629, −4.86331126143843380489271123861, −4.07081809185426965818807253880, −2.66044924043062550510326371169, −1.02635586464331361319313148619, 0, 1.52572226427085762416301879395, 3.07123768203484336667637956924, 4.01087434653002081589728224025, 5.16363537316703180111373355706, 5.91697028629810257829481358012, 6.35966648052312360341328558092, 6.97600020581979586665913916341, 8.274252989945059679740110061854, 9.048816059028418752884976070378

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