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Properties

Label 2-1792-224.69-c0-0-1
Degree $2$
Conductor $1792$
Sign $0.980 - 0.195i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.707 + 1.70i)29-s + (−0.707 − 0.292i)37-s + (−0.292 + 0.707i)43-s + 1.00i·49-s + (0.292 − 0.707i)53-s − 1.00·63-s + (0.292 + 0.707i)67-s + (1.70 − 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.707 + 1.70i)29-s + (−0.707 − 0.292i)37-s + (−0.292 + 0.707i)43-s + 1.00i·49-s + (0.292 − 0.707i)53-s − 1.00·63-s + (0.292 + 0.707i)67-s + (1.70 − 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.239284255\)
\(L(\frac12)\) \(\approx\) \(1.239284255\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - T^{2} \)
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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

−9.113887543703897793157731824346, −8.662143040434386839256474904162, −8.261420061511199060990440630327, −7.08711334295617982388325631531, −6.21576793495913288672666938555, −5.37455811743892281392160070934, −4.82380898619976756580846364012, −3.40043955131161434587863848562, −2.69214100945312238063006563296, −1.30402428085307831129897323018, 1.21016810437083424934966732154, 2.42588627887330290458188613574, 3.70794689649359819933394860728, 4.47813848181856758167666029695, 5.25395544587937548126499269955, 6.45542394977899089445833295075, 7.03167469430514836310008393642, 7.82568194844155562066348528742, 8.697089358431891805090083378100, 9.479559546712204135502919089780

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