↑

Properties

Label 1-1792-1792.1013-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.810 - 0.585i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.412 + 0.910i)3-s + (−0.999 + 0.0327i)5-s + (−0.659 − 0.751i)9-s + (0.162 + 0.986i)11-s + (0.881 − 0.471i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.227 − 0.973i)19-s + (−0.0654 + 0.997i)23-s + (0.997 − 0.0654i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.683 − 0.729i)37-s + ⋯
L(s)  = 1  + (−0.412 + 0.910i)3-s + (−0.999 + 0.0327i)5-s + (−0.659 − 0.751i)9-s + (0.162 + 0.986i)11-s + (0.881 − 0.471i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.227 − 0.973i)19-s + (−0.0654 + 0.997i)23-s + (0.997 − 0.0654i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (−0.965 + 0.258i)31-s + (−0.965 − 0.258i)33-s + (0.683 − 0.729i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.810 - 0.585i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8322101614 - 0.2692458565i\)
\(L(\frac12)\) \(\approx\) \(0.8322101614 - 0.2692458565i\)
\(L(1)\) \(\approx\) \(0.7388499406 + 0.1973747927i\)
\(L(1)\) \(\approx\) \(0.7388499406 + 0.1973747927i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.412 + 0.910i)T \)
5 \( 1 + (-0.999 + 0.0327i)T \)
11 \( 1 + (0.162 + 0.986i)T \)
13 \( 1 + (0.881 - 0.471i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.227 - 0.973i)T \)
23 \( 1 + (-0.0654 + 0.997i)T \)
29 \( 1 + (-0.634 + 0.773i)T \)
31 \( 1 + (-0.965 + 0.258i)T \)
37 \( 1 + (0.683 - 0.729i)T \)
41 \( 1 + (0.555 - 0.831i)T \)
43 \( 1 + (0.995 + 0.0980i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (0.986 - 0.162i)T \)
59 \( 1 + (-0.849 + 0.528i)T \)
61 \( 1 + (-0.812 + 0.582i)T \)
67 \( 1 + (0.910 + 0.412i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (-0.946 - 0.321i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (-0.290 + 0.956i)T \)
89 \( 1 + (-0.896 - 0.442i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.98993759124029647511670718533, −18.98569648298100376242968991173, −18.804395872215394225705547711920, −18.21478876529321296131853289152, −16.88633540129072308476928600325, −16.55023312204678436001388432799, −15.93960298127068494722806556282, −14.71296565453823860820475057000, −14.19931779051820606793577391540, −13.27362482309921788006869502268, −12.56280019328022247740274923735, −11.865676750178101388877969297902, −11.21253152763471086531972994250, −10.702687899022800311675224769778, −9.39794972536195300383978331716, −8.30181379934162811373754609282, −8.04229362460973207609403398905, −7.14400637633544801033644534255, −6.172877319565249751940261931850, −5.744212333692040121954676418338, −4.47975601237775388209427024478, −3.64728934264358787823388097511, −2.77381425812476280526196483211, −1.48595663843138079213344179201, −0.7436993516782651142706990939, 0.24645354404396304314741199855, 1.35495411045188404917214322119, 2.87470594600618000682639415504, 3.70494922643303119007931886563, 4.23248594279233909934029570972, 5.209235277022257487762067539081, 5.867565797083406204769861129888, 7.077343413176685210734263866964, 7.64184303549012259956556066360, 8.77231585023988442133254736057, 9.302299124144666084736248492423, 10.321854311211439090242531655855, 10.94453587335518922649945511510, 11.582652042635472849376501905432, 12.36592623019617669126940467678, 13.05561755618995440524203680878, 14.34167828238018525318474289982, 14.97359909746957149720458812183, 15.53024333089584750009402803522, 16.18010113180285139224327794441, 16.83668383086971180761247333384, 17.77846676958263602130612079604, 18.28569173846530004797161959914, 19.47144945927165615885579953463, 19.96940341210804173070803169931

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