Properties

Label 2-1792-1792.573-c0-0-0
Degree $2$
Conductor $1792$
Sign $0.757 + 0.653i$
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  + (0.881 − 0.471i)2-s + (0.555 − 0.831i)4-s + (0.995 − 0.0980i)7-s + (0.0980 − 0.995i)8-s + (−0.773 + 0.634i)9-s + (1.27 + 1.15i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (1.66 + 0.416i)22-s + (−1.11 − 0.598i)23-s + (−0.290 + 0.956i)25-s + (0.471 − 0.881i)28-s + (0.0788 − 1.60i)29-s + (−0.773 − 0.634i)32-s + ⋯
L(s)  = 1  + (0.881 − 0.471i)2-s + (0.555 − 0.831i)4-s + (0.995 − 0.0980i)7-s + (0.0980 − 0.995i)8-s + (−0.773 + 0.634i)9-s + (1.27 + 1.15i)11-s + (0.831 − 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (1.66 + 0.416i)22-s + (−1.11 − 0.598i)23-s + (−0.290 + 0.956i)25-s + (0.471 − 0.881i)28-s + (0.0788 − 1.60i)29-s + (−0.773 − 0.634i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.757 + 0.653i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.757 + 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.074040164\)
\(L(\frac12)\) \(\approx\) \(2.074040164\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.881 + 0.471i)T \)
7 \( 1 + (-0.995 + 0.0980i)T \)
good3 \( 1 + (0.773 - 0.634i)T^{2} \)
5 \( 1 + (0.290 - 0.956i)T^{2} \)
11 \( 1 + (-1.27 - 1.15i)T + (0.0980 + 0.995i)T^{2} \)
13 \( 1 + (-0.956 + 0.290i)T^{2} \)
17 \( 1 + (0.382 - 0.923i)T^{2} \)
19 \( 1 + (-0.471 + 0.881i)T^{2} \)
23 \( 1 + (1.11 + 0.598i)T + (0.555 + 0.831i)T^{2} \)
29 \( 1 + (-0.0788 + 1.60i)T + (-0.995 - 0.0980i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (-0.360 - 1.43i)T + (-0.881 + 0.471i)T^{2} \)
41 \( 1 + (0.831 - 0.555i)T^{2} \)
43 \( 1 + (1.88 + 0.672i)T + (0.773 + 0.634i)T^{2} \)
47 \( 1 + (-0.923 - 0.382i)T^{2} \)
53 \( 1 + (0.0887 + 1.80i)T + (-0.995 + 0.0980i)T^{2} \)
59 \( 1 + (-0.956 - 0.290i)T^{2} \)
61 \( 1 + (0.634 + 0.773i)T^{2} \)
67 \( 1 + (1.70 + 0.805i)T + (0.634 + 0.773i)T^{2} \)
71 \( 1 + (0.247 - 0.301i)T + (-0.195 - 0.980i)T^{2} \)
73 \( 1 + (-0.980 - 0.195i)T^{2} \)
79 \( 1 + (-0.373 - 1.87i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (0.881 + 0.471i)T^{2} \)
89 \( 1 + (-0.555 + 0.831i)T^{2} \)
97 \( 1 + (-0.707 + 0.707i)T^{2} \)
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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

−9.677682375087689058818392668389, −8.538730733302974213099320629397, −7.77416512185137709131543568740, −6.82229391928805315504139886849, −6.04727942269272832327624831086, −5.07110934710757056676203834440, −4.47997798109680190859908731523, −3.63644787205720967623307341980, −2.28559824886715064869792801154, −1.61061764082021150695035890797, 1.61275917254156756834825928962, 2.99832049013624461887947935216, 3.78937190016911049208239314679, 4.61798945991829713586226224983, 5.76438560231111184710615933492, 6.05995622500749312134377904163, 7.04243887484999332513701466125, 7.996135314285183176026402812226, 8.633090384274284178264840399064, 9.196114786719615629744197291818

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