↑

Properties

Label 1-19e2-361.121-r0-0-0
Degree $1$
Conductor $361$
Sign $0.377 - 0.925i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.821 − 0.569i)2-s + (0.851 − 0.523i)3-s + (0.350 + 0.936i)4-s + (−0.754 + 0.656i)5-s + (−0.998 − 0.0550i)6-s + (−0.986 + 0.164i)7-s + (0.245 − 0.969i)8-s + (0.451 − 0.892i)9-s + (0.993 − 0.110i)10-s + (0.546 − 0.837i)11-s + (0.789 + 0.614i)12-s + (0.0275 + 0.999i)13-s + (0.904 + 0.426i)14-s + (−0.298 + 0.954i)15-s + (−0.754 + 0.656i)16-s + (0.350 − 0.936i)17-s + ⋯
L(s)  = 1  + (−0.821 − 0.569i)2-s + (0.851 − 0.523i)3-s + (0.350 + 0.936i)4-s + (−0.754 + 0.656i)5-s + (−0.998 − 0.0550i)6-s + (−0.986 + 0.164i)7-s + (0.245 − 0.969i)8-s + (0.451 − 0.892i)9-s + (0.993 − 0.110i)10-s + (0.546 − 0.837i)11-s + (0.789 + 0.614i)12-s + (0.0275 + 0.999i)13-s + (0.904 + 0.426i)14-s + (−0.298 + 0.954i)15-s + (−0.754 + 0.656i)16-s + (0.350 − 0.936i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.377 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.377 - 0.925i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.377 - 0.925i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7771220379 - 0.5223188731i\)
\(L(\frac12)\) \(\approx\) \(0.7771220379 - 0.5223188731i\)
\(L(1)\) \(\approx\) \(0.7819530779 - 0.2901521261i\)
\(L(1)\) \(\approx\) \(0.7819530779 - 0.2901521261i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.821 - 0.569i)T \)
3 \( 1 + (0.851 - 0.523i)T \)
5 \( 1 + (-0.754 + 0.656i)T \)
7 \( 1 + (-0.986 + 0.164i)T \)
11 \( 1 + (0.546 - 0.837i)T \)
13 \( 1 + (0.0275 + 0.999i)T \)
17 \( 1 + (0.350 - 0.936i)T \)
23 \( 1 + (0.851 + 0.523i)T \)
29 \( 1 + (0.635 + 0.771i)T \)
31 \( 1 + (-0.0825 - 0.996i)T \)
37 \( 1 + (0.546 - 0.837i)T \)
41 \( 1 + (0.975 - 0.218i)T \)
43 \( 1 + (0.993 + 0.110i)T \)
47 \( 1 + (-0.998 - 0.0550i)T \)
53 \( 1 + (-0.998 - 0.0550i)T \)
59 \( 1 + (0.975 - 0.218i)T \)
61 \( 1 + (0.716 - 0.697i)T \)
67 \( 1 + (-0.962 + 0.272i)T \)
71 \( 1 + (0.716 + 0.697i)T \)
73 \( 1 + (0.350 - 0.936i)T \)
79 \( 1 + (0.993 + 0.110i)T \)
83 \( 1 + (0.945 - 0.324i)T \)
89 \( 1 + (0.350 + 0.936i)T \)
97 \( 1 + (-0.962 - 0.272i)T \)
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

−25.25404574186164904451285677315, −24.32398872347197027553604745605, −23.18108360952989191335448848282, −22.53385891869712266455113639120, −20.97728028432208130986251318080, −20.182512243412527887601664942925, −19.52267045141252579213005432772, −19.09995317180559415045160954577, −17.62589443346599938640426756465, −16.69720740189092419385650595731, −15.97308724806374175025156246561, −15.23284421324942833028041066918, −14.597244072153175674181807650128, −13.205968669122176918742647656, −12.34890557430156148245413464892, −10.81413323097033028237146802115, −9.96313721557929010906506418427, −9.20610320417994334319278565899, −8.31058367779702692468832095445, −7.58010536599646658316004710356, −6.48188621690753536426373204348, −5.06622043663264181243085172877, −4.00899524209238992037736637382, −2.78832961235006816280261921591, −1.151446838858729308061268338538, 0.839364202426435559999244665878, 2.43901457434917116540755005232, 3.24140838894643312663782050150, 3.96962766125138481129690100945, 6.42757831744751900372419823178, 7.07710811692292960957845324092, 7.97219763263858196438693730864, 9.11230038543682132260849969922, 9.53991404804802578447243141928, 10.98630830822734025190274084491, 11.770187837378623685075350940084, 12.64218120706322118636998320146, 13.68376265840332291875441844519, 14.629164792602369962202282433556, 15.900849049736400308688608962686, 16.41943582209072994230291784039, 17.8506714957379823376115726229, 18.84882104858196512902458588529, 19.16587584602346615885845317014, 19.73466126988684548380822339740, 20.82075823168642364412703395427, 21.75454166082117758124624073858, 22.68918357432972785977213969253, 23.78988405130213434784813366582, 24.88162850737404134080300724567

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