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Properties

Label 2-819-91.9-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.900 - 0.434i$
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  + (−0.134 − 0.232i)2-s + (0.964 − 1.66i)4-s + (−1.28 + 2.21i)5-s + (0.773 + 2.53i)7-s − 1.05·8-s + 0.686·10-s − 3.94·11-s + (−3.15 + 1.74i)13-s + (0.483 − 0.518i)14-s + (−1.78 − 3.09i)16-s + (0.392 − 0.679i)17-s − 7.49·19-s + (2.46 + 4.27i)20-s + (0.529 + 0.916i)22-s + (−3.97 − 6.88i)23-s + ⋯
L(s)  = 1  + (−0.0947 − 0.164i)2-s + (0.482 − 0.834i)4-s + (−0.572 + 0.992i)5-s + (0.292 + 0.956i)7-s − 0.372·8-s + 0.217·10-s − 1.18·11-s + (−0.874 + 0.484i)13-s + (0.129 − 0.138i)14-s + (−0.446 − 0.773i)16-s + (0.0952 − 0.164i)17-s − 1.71·19-s + (0.552 + 0.956i)20-s + (0.112 + 0.195i)22-s + (−0.829 − 1.43i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.900 - 0.434i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.900 - 0.434i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0628221 + 0.274984i\)
\(L(\frac12)\) \(\approx\) \(0.0628221 + 0.274984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.773 - 2.53i)T \)
13 \( 1 + (3.15 - 1.74i)T \)
good2 \( 1 + (0.134 + 0.232i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.28 - 2.21i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 3.94T + 11T^{2} \)
17 \( 1 + (-0.392 + 0.679i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 7.49T + 19T^{2} \)
23 \( 1 + (3.97 + 6.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.17 + 2.03i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.27 - 2.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 + 5.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.21 - 2.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.12 - 1.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.658 + 1.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.63 - 8.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.48 - 7.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 9.44T + 61T^{2} \)
67 \( 1 + 1.35T + 67T^{2} \)
71 \( 1 + (-6.15 - 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.384 + 0.665i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.09 - 5.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.07T + 83T^{2} \)
89 \( 1 + (-3.83 - 6.63i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 - 2.05i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58035716769575771039378781472, −10.07938523303857443975005162382, −8.910891919773933205319540817921, −8.042683891231017875904552872253, −7.04838176937198225977675783637, −6.31786938083971465354339408729, −5.38509929284682283016831249719, −4.35781079850898790315810194801, −2.62658950299594464058733565969, −2.29194886815494696517848460912, 0.12724036247836053559983253282, 2.07716585035774655158532891346, 3.46956328253061759701443117298, 4.39545769916176990706113522073, 5.25647587970643413979742658290, 6.60672556392443303164719119000, 7.62690596135082327611186767297, 8.002406836039047699947064074479, 8.672024602203511418191029707901, 9.992243941499556217655029809543

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