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Properties

Label 2-819-91.83-c1-0-21
Degree $2$
Conductor $819$
Sign $0.950 - 0.312i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.854 + 0.854i)2-s + 0.539i·4-s + (0.612 + 0.612i)5-s + (0.0148 − 2.64i)7-s + (−2.17 − 2.17i)8-s − 1.04·10-s + (1.85 + 1.85i)11-s + (−0.104 − 3.60i)13-s + (2.24 + 2.27i)14-s + 2.63·16-s + 3.04·17-s + (0.104 + 0.104i)19-s + (−0.330 + 0.330i)20-s − 3.17·22-s − 6.51i·23-s + ⋯
L(s)  = 1  + (−0.604 + 0.604i)2-s + 0.269i·4-s + (0.274 + 0.274i)5-s + (0.00559 − 0.999i)7-s + (−0.767 − 0.767i)8-s − 0.331·10-s + (0.559 + 0.559i)11-s + (−0.0289 − 0.999i)13-s + (0.600 + 0.607i)14-s + 0.657·16-s + 0.739·17-s + (0.0239 + 0.0239i)19-s + (−0.0739 + 0.0739i)20-s − 0.675·22-s − 1.35i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.950 - 0.312i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.950 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13321 + 0.181321i\)
\(L(\frac12)\) \(\approx\) \(1.13321 + 0.181321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.0148 + 2.64i)T \)
13 \( 1 + (0.104 + 3.60i)T \)
good2 \( 1 + (0.854 - 0.854i)T - 2iT^{2} \)
5 \( 1 + (-0.612 - 0.612i)T + 5iT^{2} \)
11 \( 1 + (-1.85 - 1.85i)T + 11iT^{2} \)
17 \( 1 - 3.04T + 17T^{2} \)
19 \( 1 + (-0.104 - 0.104i)T + 19iT^{2} \)
23 \( 1 + 6.51iT - 23T^{2} \)
29 \( 1 - 3.78T + 29T^{2} \)
31 \( 1 + (-6.77 - 6.77i)T + 31iT^{2} \)
37 \( 1 + (2.02 + 2.02i)T + 37iT^{2} \)
41 \( 1 + (-2.27 - 2.27i)T + 41iT^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + (5.21 - 5.21i)T - 47iT^{2} \)
53 \( 1 + 3.43T + 53T^{2} \)
59 \( 1 + (-9.15 + 9.15i)T - 59iT^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 + (1.04 - 1.04i)T - 67iT^{2} \)
71 \( 1 + (-4.10 + 4.10i)T - 71iT^{2} \)
73 \( 1 + (-6.92 + 6.92i)T - 73iT^{2} \)
79 \( 1 - 17.5T + 79T^{2} \)
83 \( 1 + (-10.5 - 10.5i)T + 83iT^{2} \)
89 \( 1 + (3.39 - 3.39i)T - 89iT^{2} \)
97 \( 1 + (-4.44 - 4.44i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.18492625571280479798663314284, −9.454266188195070279519151208673, −8.283109716430459366071433871953, −7.87591628127609151398564806157, −6.75111371721880976026727932483, −6.44240492212797178791876650719, −4.93184881233845760186360739651, −3.83264272362422110773379319965, −2.79123363959967891980021619094, −0.836127034051103610379332430558, 1.22072391266504647233698009534, 2.25125755686320656244593216155, 3.46940820352640853567519961954, 4.99720823739606320939178813565, 5.78459612020759650167544651343, 6.54925284097689898869820938830, 7.985385128982500995723381238802, 8.801711633146809311373553051768, 9.472486930988844716811782291313, 9.891489748692054687757304049318

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