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Properties

Label 1-19e2-361.72-r1-0-0
Degree $1$
Conductor $361$
Sign $0.618 - 0.786i$
Analytic cond. $38.7948$
Root an. cond. $38.7948$
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  + (0.435 − 0.900i)2-s + (0.263 + 0.964i)3-s + (−0.621 − 0.783i)4-s + (−0.729 − 0.684i)5-s + (0.983 + 0.182i)6-s + (0.0275 + 0.999i)7-s + (−0.975 + 0.218i)8-s + (−0.861 + 0.507i)9-s + (−0.933 + 0.359i)10-s + (0.350 − 0.936i)11-s + (0.592 − 0.805i)12-s + (−0.418 + 0.908i)13-s + (0.912 + 0.410i)14-s + (0.467 − 0.883i)15-s + (−0.227 + 0.973i)16-s + (−0.367 − 0.929i)17-s + ⋯
L(s)  = 1  + (0.435 − 0.900i)2-s + (0.263 + 0.964i)3-s + (−0.621 − 0.783i)4-s + (−0.729 − 0.684i)5-s + (0.983 + 0.182i)6-s + (0.0275 + 0.999i)7-s + (−0.975 + 0.218i)8-s + (−0.861 + 0.507i)9-s + (−0.933 + 0.359i)10-s + (0.350 − 0.936i)11-s + (0.592 − 0.805i)12-s + (−0.418 + 0.908i)13-s + (0.912 + 0.410i)14-s + (0.467 − 0.883i)15-s + (−0.227 + 0.973i)16-s + (−0.367 − 0.929i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.618 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.618 - 0.786i$
Analytic conductor: \(38.7948\)
Root analytic conductor: \(38.7948\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (1:\ ),\ 0.618 - 0.786i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.684881454 - 0.8186479329i\)
\(L(\frac12)\) \(\approx\) \(1.684881454 - 0.8186479329i\)
\(L(1)\) \(\approx\) \(1.126526105 - 0.3181898696i\)
\(L(1)\) \(\approx\) \(1.126526105 - 0.3181898696i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.435 - 0.900i)T \)
3 \( 1 + (0.263 + 0.964i)T \)
5 \( 1 + (-0.729 - 0.684i)T \)
7 \( 1 + (0.0275 + 0.999i)T \)
11 \( 1 + (0.350 - 0.936i)T \)
13 \( 1 + (-0.418 + 0.908i)T \)
17 \( 1 + (-0.367 - 0.929i)T \)
23 \( 1 + (0.967 - 0.254i)T \)
29 \( 1 + (0.979 - 0.200i)T \)
31 \( 1 + (0.962 - 0.272i)T \)
37 \( 1 + (0.986 - 0.164i)T \)
41 \( 1 + (0.951 + 0.307i)T \)
43 \( 1 + (-0.155 - 0.987i)T \)
47 \( 1 + (-0.649 + 0.760i)T \)
53 \( 1 + (0.333 + 0.942i)T \)
59 \( 1 + (-0.209 - 0.977i)T \)
61 \( 1 + (-0.842 - 0.539i)T \)
67 \( 1 + (0.991 + 0.128i)T \)
71 \( 1 + (0.842 - 0.539i)T \)
73 \( 1 + (0.989 + 0.146i)T \)
79 \( 1 + (0.777 - 0.628i)T \)
83 \( 1 + (0.451 - 0.892i)T \)
89 \( 1 + (-0.989 + 0.146i)T \)
97 \( 1 + (0.991 - 0.128i)T \)
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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

−24.56623701540944291132305834970, −23.70231662344816191060891819574, −22.9422305309409891572624125718, −22.68994828822150769968167207292, −21.2145015463555136401527900809, −19.88101404187648136863541708671, −19.51167443759334030366764777039, −18.10936097525073194450264866209, −17.5900006021725429103761339342, −16.745948051564338125267929590477, −15.32939345381061913669678946774, −14.81886745216537891317951162423, −13.97673875383192512679669464046, −12.97974393014678995188324143441, −12.33424276672206793232031915601, −11.19170928645802219235283432442, −9.902770680048915945129717195120, −8.39777934512047929366242132731, −7.70345994889200761595522523484, −6.97745152225105312912523981717, −6.34096892317540277347923729415, −4.76885292453383031327158512564, −3.71712221830111078047546419427, −2.70312539915284389326661872926, −0.80466701630539282122496744923, 0.65860567497577048519888264405, 2.41217669332751646146895267799, 3.30611472840788897295602740041, 4.48770203531716964973781174031, 4.99708375360308091720932219440, 6.217742774938936196467496498369, 8.215375453293684319540426653158, 9.07625903450656482199655008536, 9.49844588808037249982724256438, 11.03015145972155944982558507268, 11.57354477614724963400713417339, 12.354071520919667806177146889, 13.61518905624625853697844057016, 14.45844492612646149716670384684, 15.40867987074178674639026906163, 16.100054214493760652887375364462, 17.14298243157793258794285813184, 18.65543986945174404820544183264, 19.32938571112958815029034379019, 20.05294795015563047822222410364, 21.09831041910521253774736178817, 21.497226219996508373728617093504, 22.405687499623569018466560872827, 23.221331600489328474400437221195, 24.38911018403410250303215150175

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