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Properties

Label 1-19e2-361.182-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

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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: $\chi_{361} (182, \cdot )$
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 \( 1 + (-0.962 + 0.272i)T \)
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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

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

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