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Properties

Label 1-1792-1792.3-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.810 + 0.585i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
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

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.810 + 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9657377648 + 0.3124461869i\)
\(L(\frac12)\) \(\approx\) \(0.9657377648 + 0.3124461869i\)
\(L(1)\) \(\approx\) \(0.9035203193 - 0.1058366597i\)
\(L(1)\) \(\approx\) \(0.9035203193 - 0.1058366597i\)

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

−20.30112529419810074442516081652, −19.741919127192282627450918906275, −18.74901118888989172343989466115, −17.78447608622214793553605366141, −17.16035092233012783836106588455, −16.08501861724330188944158674971, −15.75929715883732231304752758744, −15.222913732760270681444940935955, −14.3536497767324000218805756302, −13.365984948624407371452070429299, −12.7891334599083822247134747505, −11.73499613576802937476139028525, −11.21978210333408892483234247005, −10.04123914219606599204340596216, −9.80243899865240945634242191692, −8.64842562300667029219363631110, −8.07638662578048545902363555708, −7.495559494149164211354133352199, −6.09820945297942625503982803097, −5.19926972349649667631467154972, −4.53755051722868901965505516508, −3.85306432125046064328230079583, −2.942891078321673673789704919324, −1.95975082935874472934424045095, −0.41169450383400251100384861545, 0.92688401964815232326608859384, 2.13129641724604380615621016976, 2.91543588769907219579624107292, 3.65719210877693461970341078635, 4.6799258154938975153103543267, 5.94303505546556113714784617767, 6.62102404542395522471891446058, 7.26213345084764524570447577784, 8.11276373411793857076514454174, 8.643063528624629693135731184736, 9.60295087329874841280409328373, 10.78965109017158082648151903909, 11.404751690498840849579271238071, 11.9417423905374151301741387352, 12.91553728449514559834029183998, 13.75313175690417772452417572453, 14.11082701735944698594006644034, 15.06432255034286235807548578904, 15.842578216177673573027086709996, 16.45231959997381506630604590007, 17.68741907754387222123920810338, 18.20979520044495827344942486717, 18.80382651074936888604092270470, 19.48793922261129089318549248936, 20.0129194664389386061078136458

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