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Properties

Label 2-819-91.9-c1-0-43
Degree $2$
Conductor $819$
Sign $-0.958 - 0.286i$
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  + (−0.379 − 0.656i)2-s + (0.712 − 1.23i)4-s + (0.357 − 0.619i)5-s + (−1.32 − 2.29i)7-s − 2.59·8-s − 0.542·10-s − 4.48·11-s + (3.26 − 1.52i)13-s + (−1.00 + 1.73i)14-s + (−0.439 − 0.761i)16-s + (−1.88 + 3.26i)17-s − 5.92·19-s + (−0.509 − 0.883i)20-s + (1.70 + 2.94i)22-s + (−0.465 − 0.806i)23-s + ⋯
L(s)  = 1  + (−0.268 − 0.464i)2-s + (0.356 − 0.616i)4-s + (0.160 − 0.277i)5-s + (−0.499 − 0.866i)7-s − 0.918·8-s − 0.171·10-s − 1.35·11-s + (0.905 − 0.424i)13-s + (−0.268 + 0.464i)14-s + (−0.109 − 0.190i)16-s + (−0.457 + 0.792i)17-s − 1.35·19-s + (−0.114 − 0.197i)20-s + (0.362 + 0.628i)22-s + (−0.0970 − 0.168i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.958 - 0.286i$
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: \((2,\ 819,\ (\ :1/2),\ -0.958 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101696 + 0.694426i\)
\(L(\frac12)\) \(\approx\) \(0.101696 + 0.694426i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 + 2.29i)T \)
13 \( 1 + (-3.26 + 1.52i)T \)
good2 \( 1 + (0.379 + 0.656i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.357 + 0.619i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.48T + 11T^{2} \)
17 \( 1 + (1.88 - 3.26i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 + (0.465 + 0.806i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.12 + 1.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.191 - 0.331i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.328 - 0.569i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.29 + 3.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.50 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.18 + 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.21 - 2.10i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.80 + 4.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 1.99T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (2.14 + 3.71i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.34 + 2.32i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.75 + 4.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 + (6.05 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.79 + 3.11i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12581601347823465080745862424, −8.989311093531308106737394064792, −8.266641044721201231975118399591, −7.11192730694748459266651784275, −6.22295278690291866314692230357, −5.45967162533991397470345985603, −4.20005911105009987405445213141, −3.00456894368585080114666854234, −1.80064115270935418277348550727, −0.34576097113995388115049028433, 2.37735001493055377379015418227, 2.99940416868583670911571795987, 4.41877266137405239088444422883, 5.76122701628164151672410111524, 6.41094414200895662901685186787, 7.21416771151656821545037475556, 8.311472500288796773821662889704, 8.718235625076509331051502626347, 9.710268093013518013443124509149, 10.75781839088105796703070933537

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