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Properties

Label 1-1792-1792.1467-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.156 + 0.987i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.973 − 0.227i)3-s + (0.812 + 0.582i)5-s + (0.896 − 0.442i)9-s + (0.0327 + 0.999i)11-s + (0.995 − 0.0980i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.935 + 0.352i)19-s + (−0.946 − 0.321i)23-s + (0.321 + 0.946i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.986 + 0.162i)37-s + ⋯
L(s)  = 1  + (0.973 − 0.227i)3-s + (0.812 + 0.582i)5-s + (0.896 − 0.442i)9-s + (0.0327 + 0.999i)11-s + (0.995 − 0.0980i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (−0.935 + 0.352i)19-s + (−0.946 − 0.321i)23-s + (0.321 + 0.946i)25-s + (0.773 − 0.634i)27-s + (0.471 + 0.881i)29-s + (−0.258 + 0.965i)31-s + (0.258 + 0.965i)33-s + (−0.986 + 0.162i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1467, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.414332889 + 2.828075663i\)
\(L(\frac12)\) \(\approx\) \(2.414332889 + 2.828075663i\)
\(L(1)\) \(\approx\) \(1.697274142 + 0.4164296238i\)
\(L(1)\) \(\approx\) \(1.697274142 + 0.4164296238i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.973 - 0.227i)T \)
5 \( 1 + (0.812 + 0.582i)T \)
11 \( 1 + (0.0327 + 0.999i)T \)
13 \( 1 + (0.995 - 0.0980i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (-0.935 + 0.352i)T \)
23 \( 1 + (-0.946 - 0.321i)T \)
29 \( 1 + (0.471 + 0.881i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (-0.986 + 0.162i)T \)
41 \( 1 + (-0.980 + 0.195i)T \)
43 \( 1 + (0.290 + 0.956i)T \)
47 \( 1 + (-0.991 - 0.130i)T \)
53 \( 1 + (-0.999 + 0.0327i)T \)
59 \( 1 + (0.412 + 0.910i)T \)
61 \( 1 + (0.729 - 0.683i)T \)
67 \( 1 + (0.227 + 0.973i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (0.997 + 0.0654i)T \)
79 \( 1 + (-0.608 - 0.793i)T \)
83 \( 1 + (-0.634 + 0.773i)T \)
89 \( 1 + (-0.751 - 0.659i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.9179740042166178600846315960, −19.025098618782175821205250091483, −18.58587186637769745258164187484, −17.6028387996752351036577889134, −16.75695959886648943012558881274, −16.07611946177925747642545976162, −15.49190010233658992583096354741, −14.41223162921581999077431480734, −13.77586077378599431245934165534, −13.43577023089386891629489320976, −12.57768158679615459707362829872, −11.56738106056872547571822371217, −10.59733881124351810035014228927, −9.87157617862820327668438264104, −9.17101795504757140694268275876, −8.439927715932251750739709454655, −7.988076407245154528704813983679, −6.70903180862061117805993761090, −5.92081745754775451776376684600, −5.09211721023068602549613696926, −4.05921569150201685108808519034, −3.35905202610220775144606601381, −2.32653604306580995915017089335, −1.56483095969206151789163010547, −0.49620868042980741500503943948, 1.484852438575746689997112226696, 1.77759286261470997168859493171, 2.91952304793615529381161923678, 3.60072228567672881657488704305, 4.55650116460077245887767438079, 5.733534586195885700099294312505, 6.581348173183071949036221852637, 7.13181470548305121210676323181, 8.26762664281568142840015843075, 8.67213571534504178477514024707, 9.929651596623265154677732105300, 10.08261267658429327137318923638, 11.04562431358636385224657915111, 12.3857479804846924732456034947, 12.77342873610842425388617538809, 13.677214349486517259464667474222, 14.38287847849934007354592061137, 14.7722420303628189583015172917, 15.63598502794789159788127465374, 16.49058002373127972405543364401, 17.58919245791006056462571203416, 18.05330492741998140279744072133, 18.757101081076228866142561740168, 19.44591354625945762378690495066, 20.29604605313386012462834810951

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