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Properties

Label 2-819-91.5-c1-0-12
Degree $2$
Conductor $819$
Sign $-0.218 + 0.975i$
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  + (−0.697 − 2.60i)2-s + (−4.55 + 2.63i)4-s + (−2.44 + 0.654i)5-s + (0.722 − 2.54i)7-s + (6.22 + 6.22i)8-s + (3.40 + 5.89i)10-s + (−0.557 + 2.08i)11-s + (1.44 + 3.30i)13-s + (−7.13 − 0.104i)14-s + (6.59 − 11.4i)16-s + (0.700 + 1.21i)17-s + (2.02 − 0.541i)19-s + (9.40 − 9.40i)20-s + 5.80·22-s + (1.13 + 0.657i)23-s + ⋯
L(s)  = 1  + (−0.493 − 1.84i)2-s + (−2.27 + 1.31i)4-s + (−1.09 + 0.292i)5-s + (0.272 − 0.962i)7-s + (2.19 + 2.19i)8-s + (1.07 + 1.86i)10-s + (−0.168 + 0.627i)11-s + (0.401 + 0.915i)13-s + (−1.90 − 0.0279i)14-s + (1.64 − 2.85i)16-s + (0.169 + 0.294i)17-s + (0.463 − 0.124i)19-s + (2.10 − 2.10i)20-s + 1.23·22-s + (0.237 + 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.218 + 0.975i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.218 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493880 - 0.616422i\)
\(L(\frac12)\) \(\approx\) \(0.493880 - 0.616422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.722 + 2.54i)T \)
13 \( 1 + (-1.44 - 3.30i)T \)
good2 \( 1 + (0.697 + 2.60i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (2.44 - 0.654i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.557 - 2.08i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.700 - 1.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.02 + 0.541i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.13 - 0.657i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + (-1.88 + 7.03i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.20 - 0.591i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.69 - 2.69i)T + 41iT^{2} \)
43 \( 1 + 0.437iT - 43T^{2} \)
47 \( 1 + (2.07 + 7.74i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.54 - 2.02i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.57 - 3.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.548 + 0.146i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.7 + 10.7i)T - 71iT^{2} \)
73 \( 1 + (-11.8 - 3.18i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.19 - 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.82 - 3.82i)T + 83iT^{2} \)
89 \( 1 + (-0.0134 - 0.0501i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-9.43 - 9.43i)T + 97iT^{2} \)
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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

−10.14176219492156380737022315581, −9.510599842114378937844422758723, −8.409715963440527663095910296198, −7.80427638125178558417593774954, −6.90304780304506747830782640434, −4.88079140570925287282109395882, −4.05373637800564064335254020041, −3.52277640947422917258568547352, −2.16547732264623979692051883387, −0.843510488872173291962230562027, 0.75058671985532086382364319515, 3.33271786902661207246867973833, 4.67061279681972392696574873029, 5.37998595498701140681633834058, 6.16337214786738065133476042232, 7.18801013469519263226206385277, 8.112556911988790679121484662123, 8.358258358860482078617736035841, 9.131192887757460871536627327552, 10.15583262165130439559609426432

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