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Properties

Label 2-819-91.5-c1-0-5
Degree $2$
Conductor $819$
Sign $-0.478 + 0.878i$
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  + (0.639 + 2.38i)2-s + (−3.56 + 2.05i)4-s + (1.58 − 0.423i)5-s + (−2.54 − 0.716i)7-s + (−3.69 − 3.69i)8-s + (2.02 + 3.50i)10-s + (−1.48 + 5.55i)11-s + (−3.57 + 0.473i)13-s + (0.0809 − 6.54i)14-s + (2.34 − 4.06i)16-s + (−0.991 − 1.71i)17-s + (−0.918 + 0.246i)19-s + (−4.76 + 4.76i)20-s − 14.2·22-s + (3.06 + 1.77i)23-s + ⋯
L(s)  = 1  + (0.452 + 1.68i)2-s + (−1.78 + 1.02i)4-s + (0.707 − 0.189i)5-s + (−0.962 − 0.270i)7-s + (−1.30 − 1.30i)8-s + (0.640 + 1.10i)10-s + (−0.448 + 1.67i)11-s + (−0.991 + 0.131i)13-s + (0.0216 − 1.74i)14-s + (0.587 − 1.01i)16-s + (−0.240 − 0.416i)17-s + (−0.210 + 0.0564i)19-s + (−1.06 + 1.06i)20-s − 3.03·22-s + (0.639 + 0.369i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.478 + 0.878i$
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: \((2,\ 819,\ (\ :1/2),\ -0.478 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498875 - 0.839855i\)
\(L(\frac12)\) \(\approx\) \(0.498875 - 0.839855i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.716i)T \)
13 \( 1 + (3.57 - 0.473i)T \)
good2 \( 1 + (-0.639 - 2.38i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.58 + 0.423i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.48 - 5.55i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.991 + 1.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.918 - 0.246i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.06 - 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.83T + 29T^{2} \)
31 \( 1 + (-1.16 + 4.33i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.73 - 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.02 + 4.02i)T + 41iT^{2} \)
43 \( 1 - 5.30iT - 43T^{2} \)
47 \( 1 + (0.120 + 0.448i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.31 - 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-11.5 - 3.10i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.38 + 2.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.87 - 1.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.84 + 4.84i)T - 71iT^{2} \)
73 \( 1 + (4.24 + 1.13i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.08 - 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 - 11.5i)T + 83iT^{2} \)
89 \( 1 + (-0.941 - 3.51i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.09 - 7.09i)T + 97iT^{2} \)
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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

−10.39283280617742104361208567708, −9.556929019842485505878291092529, −9.203096304690748363052630662610, −7.81842311824244139910337927270, −7.16461459058474165213692866861, −6.63048765949501078232980184391, −5.55411600780246311871486288811, −4.90940572303471540185906335976, −3.95649679838678195944704212459, −2.34923564908063941819927273484, 0.38803665946075818820418717706, 2.11026284750503325837891475940, 2.95381984495754692697767562887, 3.68688253215670094720701447114, 5.08633665550920279088970429947, 5.79379353967339706933840757639, 6.82424741307507937403487607427, 8.454801762309804864254112550286, 9.148778421396774964954232980880, 10.09939786353055003920448742761

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