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Properties

Label 2-819-117.2-c1-0-33
Degree $2$
Conductor $819$
Sign $-0.693 - 0.720i$
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  + (1.92 + 1.92i)2-s + (0.763 − 1.55i)3-s + 5.40i·4-s + (−0.462 + 1.72i)5-s + (4.46 − 1.52i)6-s + (−0.258 + 0.965i)7-s + (−6.54 + 6.54i)8-s + (−1.83 − 2.37i)9-s + (−4.21 + 2.43i)10-s + (1.36 − 1.36i)11-s + (8.39 + 4.12i)12-s + (3.20 + 1.65i)13-s + (−2.35 + 1.36i)14-s + (2.33 + 2.03i)15-s − 14.3·16-s + (−2.56 + 4.44i)17-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)2-s + (0.440 − 0.897i)3-s + 2.70i·4-s + (−0.206 + 0.771i)5-s + (1.82 − 0.621i)6-s + (−0.0978 + 0.365i)7-s + (−2.31 + 2.31i)8-s + (−0.611 − 0.791i)9-s + (−1.33 + 0.768i)10-s + (0.411 − 0.411i)11-s + (2.42 + 1.19i)12-s + (0.888 + 0.458i)13-s + (−0.629 + 0.363i)14-s + (0.601 + 0.526i)15-s − 3.59·16-s + (−0.622 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.693 - 0.720i$
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: \((2,\ 819,\ (\ :1/2),\ -0.693 - 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30359 + 3.06269i\)
\(L(\frac12)\) \(\approx\) \(1.30359 + 3.06269i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.763 + 1.55i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-3.20 - 1.65i)T \)
good2 \( 1 + (-1.92 - 1.92i)T + 2iT^{2} \)
5 \( 1 + (0.462 - 1.72i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.36 + 1.36i)T - 11iT^{2} \)
17 \( 1 + (2.56 - 4.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.342 - 1.27i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.59 + 2.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.60iT - 29T^{2} \)
31 \( 1 + (1.86 + 0.500i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.0591 + 0.220i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.48 + 0.665i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.52 + 3.76i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.72 - 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 + (-10.7 + 10.7i)T - 59iT^{2} \)
61 \( 1 + (-0.309 - 0.535i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.542 + 2.02i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.24 - 0.870i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.29 + 3.29i)T + 73iT^{2} \)
79 \( 1 + (5.37 - 9.30i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.3 + 2.77i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-5.37 - 1.44i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.2 - 3.82i)T + (84.0 + 48.5i)T^{2} \)
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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

−11.01761683059432185223604400116, −9.119994305782449981560266232460, −8.419358846314542052106379009184, −7.75213537462024933515457598120, −6.72511623371015325803345552694, −6.39618858074498401138252658068, −5.64423572757455845175937296959, −4.11500974871420815558103134356, −3.45119326663448493134602443913, −2.35444622829197886077742987489, 1.09622946870705013402356736138, 2.59520411007052100785941313163, 3.53792772452684160846024878566, 4.33649563350918108584028150010, 4.96528815469890511612844941495, 5.79815027135537944677340726614, 7.16707324992566310145716300517, 8.845349281745602551115629443151, 9.227913695320563837230718009904, 10.24275420855792668005713125962

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