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Properties

Label 1-1792-1792.1357-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: $\chi_{1792} (1357, \cdot )$
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

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

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