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Properties

Label 2-819-91.54-c1-0-22
Degree $2$
Conductor $819$
Sign $0.888 - 0.459i$
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  + (0.347 − 0.347i)2-s + 1.75i·4-s + (3.47 − 0.931i)5-s + (0.701 + 2.55i)7-s + (1.30 + 1.30i)8-s + (0.883 − 1.52i)10-s + (2.04 − 0.547i)11-s + (0.582 − 3.55i)13-s + (1.12 + 0.642i)14-s − 2.61·16-s − 1.14·17-s + (−1.18 + 4.40i)19-s + (1.63 + 6.11i)20-s + (0.519 − 0.899i)22-s − 1.48i·23-s + ⋯
L(s)  = 1  + (0.245 − 0.245i)2-s + 0.879i·4-s + (1.55 − 0.416i)5-s + (0.264 + 0.964i)7-s + (0.461 + 0.461i)8-s + (0.279 − 0.483i)10-s + (0.616 − 0.165i)11-s + (0.161 − 0.986i)13-s + (0.301 + 0.171i)14-s − 0.653·16-s − 0.277·17-s + (−0.270 + 1.01i)19-s + (0.366 + 1.36i)20-s + (0.110 − 0.191i)22-s − 0.309i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.888 - 0.459i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.888 - 0.459i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34956 + 0.571169i\)
\(L(\frac12)\) \(\approx\) \(2.34956 + 0.571169i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.701 - 2.55i)T \)
13 \( 1 + (-0.582 + 3.55i)T \)
good2 \( 1 + (-0.347 + 0.347i)T - 2iT^{2} \)
5 \( 1 + (-3.47 + 0.931i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.04 + 0.547i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 1.14T + 17T^{2} \)
19 \( 1 + (1.18 - 4.40i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 1.48iT - 23T^{2} \)
29 \( 1 + (2.75 + 4.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.56 - 5.85i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.91 + 1.91i)T + 37iT^{2} \)
41 \( 1 + (-1.21 + 4.54i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.55 + 2.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.74 - 6.51i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.74 - 3.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.08 + 8.08i)T - 59iT^{2} \)
61 \( 1 + (-8.20 + 4.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.99 - 7.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.74 + 6.52i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.46 - 1.46i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.91 + 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.06 + 5.06i)T + 83iT^{2} \)
89 \( 1 + (7.05 - 7.05i)T - 89iT^{2} \)
97 \( 1 + (13.1 - 3.52i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.27948108936043756863276059561, −9.343094643253988191924365058040, −8.673880442219759838286163472439, −8.002656320764706942051306958665, −6.64803431719743822387438409443, −5.74276779221458814876842420392, −5.11260727828364343271112479390, −3.79527459444796641948862956344, −2.58888656042929807358809844562, −1.73299369973752174619101440058, 1.31380626785640150750998757429, 2.21985104123869884208147027675, 3.97527831543360242276895916527, 4.94121109270221429122557927618, 5.83094075333883616384468018185, 6.83082394461962196969049317715, 6.95832077309657575803920420472, 8.732750991439966019811193616218, 9.612007513729557456875296284163, 9.983755662479554309281201717003

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