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Properties

Label 2-819-117.2-c1-0-42
Degree $2$
Conductor $819$
Sign $0.999 - 0.0301i$
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  + (0.338 + 0.338i)2-s + (−0.471 + 1.66i)3-s − 1.77i·4-s + (−0.0784 + 0.292i)5-s + (−0.722 + 0.404i)6-s + (0.258 − 0.965i)7-s + (1.27 − 1.27i)8-s + (−2.55 − 1.57i)9-s + (−0.125 + 0.0724i)10-s + (−1.64 + 1.64i)11-s + (2.95 + 0.834i)12-s + (3.60 + 0.0133i)13-s + (0.414 − 0.239i)14-s + (−0.450 − 0.268i)15-s − 2.68·16-s + (3.07 − 5.32i)17-s + ⋯
L(s)  = 1  + (0.239 + 0.239i)2-s + (−0.272 + 0.962i)3-s − 0.885i·4-s + (−0.0350 + 0.130i)5-s + (−0.295 + 0.164i)6-s + (0.0978 − 0.365i)7-s + (0.450 − 0.450i)8-s + (−0.851 − 0.523i)9-s + (−0.0396 + 0.0229i)10-s + (−0.495 + 0.495i)11-s + (0.852 + 0.241i)12-s + (0.999 + 0.00370i)13-s + (0.110 − 0.0638i)14-s + (−0.116 − 0.0693i)15-s − 0.670·16-s + (0.745 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0301i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0301i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.999 - 0.0301i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62899 + 0.0245799i\)
\(L(\frac12)\) \(\approx\) \(1.62899 + 0.0245799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.471 - 1.66i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (-3.60 - 0.0133i)T \)
good2 \( 1 + (-0.338 - 0.338i)T + 2iT^{2} \)
5 \( 1 + (0.0784 - 0.292i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.64 - 1.64i)T - 11iT^{2} \)
17 \( 1 + (-3.07 + 5.32i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.76 - 6.60i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.43 + 7.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.39iT - 29T^{2} \)
31 \( 1 + (-2.55 - 0.685i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.979 - 3.65i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.597 + 0.160i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.74 + 3.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.42 - 5.30i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 5.20iT - 53T^{2} \)
59 \( 1 + (-4.87 + 4.87i)T - 59iT^{2} \)
61 \( 1 + (1.36 + 2.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.43 - 5.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.24 - 1.13i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (9.16 + 9.16i)T + 73iT^{2} \)
79 \( 1 + (5.94 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.19 + 2.46i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (5.90 + 1.58i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.72 - 0.462i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.25903111673086699052272727062, −9.665597116145470429474976584935, −8.718608572331613994275312950223, −7.58628771342412197840392274379, −6.52403675647050273459942928272, −5.70599547166268074940172348871, −4.90924259741394188950300724363, −4.12373225350565196222396636704, −2.86272813433297766936799225981, −0.942531490999984210332489207783, 1.28006321272220580616415780134, 2.71572943088701442865594876712, 3.52236817471755072650655782833, 4.99896015730105424640034084506, 5.79630753661012768550280121413, 6.91060216446737391517506190098, 7.64071845523035480027763938435, 8.533884793718169475077175194241, 8.951670428783059616406655929706, 10.67690967813142894172147357515

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