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Properties

Label 1-1792-1792.341-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.585 + 0.810i$
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  + (−0.352 − 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (−0.991 − 0.130i)17-s + (−0.849 − 0.528i)19-s + (0.997 + 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (−0.0980 − 0.995i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.999 − 0.0327i)37-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (−0.382 + 0.923i)15-s + (−0.991 − 0.130i)17-s + (−0.849 − 0.528i)19-s + (0.997 + 0.0654i)23-s + (0.0654 + 0.997i)25-s + (0.881 + 0.471i)27-s + (−0.0980 − 0.995i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.999 − 0.0327i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.585 + 0.810i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2242474259 - 0.1146149226i\)
\(L(\frac12)\) \(\approx\) \(-0.2242474259 - 0.1146149226i\)
\(L(1)\) \(\approx\) \(0.5600648059 - 0.4176186007i\)
\(L(1)\) \(\approx\) \(0.5600648059 - 0.4176186007i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.352 - 0.935i)T \)
5 \( 1 + (-0.729 - 0.683i)T \)
11 \( 1 + (0.582 - 0.812i)T \)
13 \( 1 + (0.290 - 0.956i)T \)
17 \( 1 + (-0.991 - 0.130i)T \)
19 \( 1 + (-0.849 - 0.528i)T \)
23 \( 1 + (0.997 + 0.0654i)T \)
29 \( 1 + (-0.0980 - 0.995i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (0.999 - 0.0327i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.634 - 0.773i)T \)
47 \( 1 + (0.793 + 0.608i)T \)
53 \( 1 + (-0.812 - 0.582i)T \)
59 \( 1 + (-0.973 - 0.227i)T \)
61 \( 1 + (0.162 - 0.986i)T \)
67 \( 1 + (-0.935 + 0.352i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.471 + 0.881i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
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

−20.51127446380195379365041410649, −20.05931236510869815552685066235, −19.221470212103316525744032663255, −18.415608496024454531545224280766, −17.68225036948742393502243657090, −16.71993045264411153848588024798, −16.36621529603094061577724000002, −15.269664488670166430576631833, −14.9211521688822430372074000837, −14.35723082849624345958953180216, −13.17573545490045628343825430742, −12.192582449417997894832210293814, −11.52720561300514476085376019620, −10.9001399992776982503006329619, −10.31508434209144367038945143109, −9.21660161210557418160902929426, −8.82899342781899186057773765611, −7.63498515113779608740573958680, −6.6788095694077199094361336970, −6.276805210259874299532813872739, −4.91114733957335511801348591133, −4.273088038998755190643247274, −3.69410471084007399355006845622, −2.68439849521342910160533094322, −1.51903004198657701040376925496, 0.07312496098083849852001813886, 0.63471700088197862968564640819, 1.59794605032368917759335577445, 2.73315123980422612208737831805, 3.69890867273666771663356389499, 4.70539742458685315039098723537, 5.52520770001422488362240279785, 6.36822198673268743639350813101, 7.12416502618864799676223092954, 8.01417942200916314493229350843, 8.60093061895953891773846333371, 9.24978129538424136636652113847, 10.814760576107753812544833947141, 11.15801430577860800222716318029, 11.92935349378958132196378861208, 12.83433528631398284790425455608, 13.16064812073434028401030420378, 13.99500640092911357675753627005, 15.11777673595674352455883642360, 15.66611162946983385551407617270, 16.72757082289574421755931697316, 17.09568394669282567154320520041, 17.89996636486891162383651196916, 18.81656618710207548063083539898, 19.33378546336066891862290088906

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