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Properties

Label 2-1792-16.5-c1-0-30
Degree $2$
Conductor $1792$
Sign $0.608 + 0.793i$
Analytic cond. $14.3091$
Root an. cond. $3.78274$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.171 − 0.171i)3-s + (−0.268 − 0.268i)5-s + i·7-s + 2.94i·9-s + (−1.84 − 1.84i)11-s + (1.63 − 1.63i)13-s − 0.0919·15-s − 7.37·17-s + (3.84 − 3.84i)19-s + (0.171 + 0.171i)21-s − 6.44i·23-s − 4.85i·25-s + (1.01 + 1.01i)27-s + (3.58 − 3.58i)29-s + 6.10·31-s + ⋯
L(s)  = 1  + (0.0988 − 0.0988i)3-s + (−0.120 − 0.120i)5-s + 0.377i·7-s + 0.980i·9-s + (−0.557 − 0.557i)11-s + (0.453 − 0.453i)13-s − 0.0237·15-s − 1.78·17-s + (0.883 − 0.883i)19-s + (0.0373 + 0.0373i)21-s − 1.34i·23-s − 0.971i·25-s + (0.195 + 0.195i)27-s + (0.666 − 0.666i)29-s + 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(14.3091\)
Root analytic conductor: \(3.78274\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.510424144\)
\(L(\frac12)\) \(\approx\) \(1.510424144\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + (-0.171 + 0.171i)T - 3iT^{2} \)
5 \( 1 + (0.268 + 0.268i)T + 5iT^{2} \)
11 \( 1 + (1.84 + 1.84i)T + 11iT^{2} \)
13 \( 1 + (-1.63 + 1.63i)T - 13iT^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
19 \( 1 + (-3.84 + 3.84i)T - 19iT^{2} \)
23 \( 1 + 6.44iT - 23T^{2} \)
29 \( 1 + (-3.58 + 3.58i)T - 29iT^{2} \)
31 \( 1 - 6.10T + 31T^{2} \)
37 \( 1 + (-7.41 - 7.41i)T + 37iT^{2} \)
41 \( 1 - 0.836iT - 41T^{2} \)
43 \( 1 + (-3.88 - 3.88i)T + 43iT^{2} \)
47 \( 1 - 6.02T + 47T^{2} \)
53 \( 1 + (-0.575 - 0.575i)T + 53iT^{2} \)
59 \( 1 + (5.33 + 5.33i)T + 59iT^{2} \)
61 \( 1 + (-0.929 + 0.929i)T - 61iT^{2} \)
67 \( 1 + (-6.21 + 6.21i)T - 67iT^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 - 3.68iT - 73T^{2} \)
79 \( 1 - 4.21T + 79T^{2} \)
83 \( 1 + (-12.0 + 12.0i)T - 83iT^{2} \)
89 \( 1 - 9.32iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.987548297586325774383323640421, −8.275390401377532814249007600275, −7.88784674208241341911397675171, −6.64029106512047514872318405697, −6.08302304255417996912300225878, −4.85737395582153274931009759347, −4.48431518777170024814531899587, −2.85622270023663880972202374945, −2.38216425611184608825775750952, −0.63519501720567193274701367711, 1.14755809294726428681470384519, 2.47920357798212152827158681942, 3.64467876212336019563906823384, 4.25015028403506772764807096505, 5.35561904828627040919669440761, 6.27643606714688145955857799420, 7.08180849720389769731347813885, 7.68505635350244872960583312908, 8.785050262511859426489956157004, 9.338786860781924512236716322891

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