Properties

Label 2-1792-1792.125-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  + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (−0.0980 − 0.995i)7-s + (0.995 + 0.0980i)8-s + (−0.634 − 0.773i)9-s + (−1.93 + 0.0951i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.997 + 1.66i)22-s + (−0.728 + 1.36i)23-s + (0.956 + 0.290i)25-s + (0.881 + 0.471i)28-s + (−0.217 + 0.197i)29-s + (−0.634 + 0.773i)32-s + ⋯
L(s)  = 1  + (−0.471 − 0.881i)2-s + (−0.555 + 0.831i)4-s + (−0.0980 − 0.995i)7-s + (0.995 + 0.0980i)8-s + (−0.634 − 0.773i)9-s + (−1.93 + 0.0951i)11-s + (−0.831 + 0.555i)14-s + (−0.382 − 0.923i)16-s + (−0.382 + 0.923i)18-s + (0.997 + 1.66i)22-s + (−0.728 + 1.36i)23-s + (0.956 + 0.290i)25-s + (0.881 + 0.471i)28-s + (−0.217 + 0.197i)29-s + (−0.634 + 0.773i)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: $\chi_{1792} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.757 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1957245563\)
\(L(\frac12)\) \(\approx\) \(0.1957245563\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.471 + 0.881i)T \)
7 \( 1 + (0.0980 + 0.995i)T \)
good3 \( 1 + (0.634 + 0.773i)T^{2} \)
5 \( 1 + (-0.956 - 0.290i)T^{2} \)
11 \( 1 + (1.93 - 0.0951i)T + (0.995 - 0.0980i)T^{2} \)
13 \( 1 + (-0.290 - 0.956i)T^{2} \)
17 \( 1 + (0.382 - 0.923i)T^{2} \)
19 \( 1 + (-0.881 - 0.471i)T^{2} \)
23 \( 1 + (0.728 - 1.36i)T + (-0.555 - 0.831i)T^{2} \)
29 \( 1 + (0.217 - 0.197i)T + (0.0980 - 0.995i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (1.71 + 1.02i)T + (0.471 + 0.881i)T^{2} \)
41 \( 1 + (-0.831 + 0.555i)T^{2} \)
43 \( 1 + (1.21 - 0.574i)T + (0.634 - 0.773i)T^{2} \)
47 \( 1 + (-0.923 - 0.382i)T^{2} \)
53 \( 1 + (0.499 + 0.452i)T + (0.0980 + 0.995i)T^{2} \)
59 \( 1 + (-0.290 + 0.956i)T^{2} \)
61 \( 1 + (-0.773 + 0.634i)T^{2} \)
67 \( 1 + (0.609 + 1.70i)T + (-0.773 + 0.634i)T^{2} \)
71 \( 1 + (0.301 + 0.247i)T + (0.195 + 0.980i)T^{2} \)
73 \( 1 + (0.980 + 0.195i)T^{2} \)
79 \( 1 + (0.113 + 0.569i)T + (-0.923 + 0.382i)T^{2} \)
83 \( 1 + (-0.471 + 0.881i)T^{2} \)
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

−9.149559483315402269449384054958, −8.225192652510965548851212470188, −7.64187021336730339124345089561, −6.87961483544832181331403216249, −5.54670155179172277668884291645, −4.77494238206770436262759687650, −3.55978015895964031716844697781, −3.06060042387548082657017742979, −1.75011280375004429612024393860, −0.16029238220348089392427428429, 2.10524724599924165294311447639, 2.94534561530824284198070873759, 4.70856423362891825247711130857, 5.25265101012469532639041256669, 5.88675528091386924780144929810, 6.81509129228228833266725071247, 7.77968902579534365473411477028, 8.521188984166169862626848306568, 8.644430980279100902216905281113, 10.03866308608532268680879917672

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