Properties

Label 1-1792-1792.1413-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.219 + 0.975i$
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  + (−0.956 − 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.831 + 0.555i)9-s + (0.471 − 0.881i)11-s + (−0.0980 − 0.995i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (−0.634 + 0.773i)19-s + (0.195 + 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 − 0.634i)37-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)3-s + (0.995 + 0.0980i)5-s + (0.831 + 0.555i)9-s + (0.471 − 0.881i)11-s + (−0.0980 − 0.995i)13-s + (−0.923 − 0.382i)15-s + (0.923 − 0.382i)17-s + (−0.634 + 0.773i)19-s + (0.195 + 0.980i)23-s + (0.980 + 0.195i)25-s + (−0.634 − 0.773i)27-s + (−0.881 + 0.471i)29-s + (−0.707 − 0.707i)31-s + (−0.707 + 0.707i)33-s + (0.773 − 0.634i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.219 + 0.975i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.219 + 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5290655534 + 0.6610464941i\)
\(L(\frac12)\) \(\approx\) \(0.5290655534 + 0.6610464941i\)
\(L(1)\) \(\approx\) \(0.8852365874 - 0.05412550474i\)
\(L(1)\) \(\approx\) \(0.8852365874 - 0.05412550474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.956 + 0.290i)T \)
5 \( 1 + (-0.995 - 0.0980i)T \)
11 \( 1 + (-0.471 + 0.881i)T \)
13 \( 1 + (0.0980 + 0.995i)T \)
17 \( 1 + (-0.923 + 0.382i)T \)
19 \( 1 + (0.634 - 0.773i)T \)
23 \( 1 + (-0.195 - 0.980i)T \)
29 \( 1 + (0.881 - 0.471i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.773 + 0.634i)T \)
41 \( 1 + (0.980 - 0.195i)T \)
43 \( 1 + (0.956 - 0.290i)T \)
47 \( 1 + (-0.382 - 0.923i)T \)
53 \( 1 + (0.881 + 0.471i)T \)
59 \( 1 + (0.0980 - 0.995i)T \)
61 \( 1 + (0.290 - 0.956i)T \)
67 \( 1 + (0.290 - 0.956i)T \)
71 \( 1 + (0.831 - 0.555i)T \)
73 \( 1 + (0.555 - 0.831i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.773 + 0.634i)T \)
89 \( 1 + (0.195 - 0.980i)T \)
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

−19.94371938849547239079678913081, −18.72540709671348137609696883804, −18.372288224069079351683548325581, −17.35065449526861369032181298445, −16.91158984060868849492261001780, −16.52601801079384056499446983629, −15.298447704792590601576198417531, −14.73333364605046966522792385498, −13.885941479481655892356795196172, −12.90275684232234853504807355615, −12.38776769466165307518475454775, −11.56765796138870224334926283897, −10.732011882305508737823219845385, −9.99869606977330867231524313087, −9.43951929381550889932437586730, −8.66364987954496171732592563576, −7.262868489029947427631063207860, −6.59807604798812785012359649353, −6.03224097001942804424334216030, −4.97497505060767540337269798356, −4.55180763469951122149348236453, −3.448123480091116397748305262234, −2.03675762998008979479677326815, −1.476195256584461997800556094222, −0.17984482100112849480853467790, 1.022754074881489903801526403551, 1.653260919100420329088876689242, 2.875520874269277388167719817764, 3.82126483934424432520510985363, 5.083842877480570511838673089398, 5.77875417238922389832927732637, 6.05741812424393659182872438536, 7.183176214465800403545051060195, 7.8791972877428229900514852508, 9.002031245933990996046895795105, 9.8670967361151496698226635525, 10.46598392099203733658967119380, 11.24158489003450529066609497172, 11.95019080027701307640797325936, 13.002268272648791282153934074252, 13.219681255123854596816586469073, 14.31240931085961203099506233010, 14.93288403405965001401361770606, 16.12238662294500659372310855329, 16.72617501483717429108784400999, 17.218657736975022352812672791176, 18.01685813304557111824701177798, 18.61139889455043332625565140989, 19.24128345716873387689718228535, 20.35384821147523472946082433140

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