Properties

Label 2-1792-448.69-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.956 + 0.290i$
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  + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (−0.216 − 0.324i)11-s + (0.707 − 0.292i)23-s + (0.923 + 0.382i)25-s + (1.08 − 1.63i)29-s + (−0.216 + 1.08i)37-s + (−0.923 + 0.617i)43-s + (−0.707 − 0.707i)49-s + (−0.923 − 1.38i)53-s + 63-s + (1.38 + 0.923i)67-s + (0.541 − 1.30i)71-s + (−0.382 + 0.0761i)77-s + (1.30 + 1.30i)79-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)7-s + (0.382 + 0.923i)9-s + (−0.216 − 0.324i)11-s + (0.707 − 0.292i)23-s + (0.923 + 0.382i)25-s + (1.08 − 1.63i)29-s + (−0.216 + 1.08i)37-s + (−0.923 + 0.617i)43-s + (−0.707 − 0.707i)49-s + (−0.923 − 1.38i)53-s + 63-s + (1.38 + 0.923i)67-s + (0.541 − 1.30i)71-s + (−0.382 + 0.0761i)77-s + (1.30 + 1.30i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232922352\)
\(L(\frac12)\) \(\approx\) \(1.232922352\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (-0.382 - 0.923i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
11 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
13 \( 1 + (-0.923 + 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T^{2} \)
43 \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2} \)
59 \( 1 + (-0.923 - 0.382i)T^{2} \)
61 \( 1 + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \)
71 \( 1 + (-0.541 + 1.30i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
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

−9.617300844588676441224855198324, −8.301505872208087438308487958892, −8.063821145896901604297040894188, −7.04198909847934107909991681506, −6.42898529256625276954164352132, −5.07975479498243589119725502495, −4.68293973275078591872219714204, −3.57737563456813007123505827355, −2.45673267128541477511201942361, −1.16337105204809241708173268299, 1.35273966865603683291178777174, 2.62283527555943498779730812947, 3.54206089392569076294207002983, 4.73793429289789566293260991401, 5.34918744240314911876443062812, 6.43072636855418341135310867958, 7.01119381271110769742713054870, 8.019079866897780880256512781554, 8.910795968755229013846123112709, 9.256605641591234590716614491715

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