Properties

Label 2-819-91.45-c1-0-1
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  + (−1.74 + 1.74i)2-s − 4.08i·4-s + (−0.130 + 0.488i)5-s + (1.09 + 2.40i)7-s + (3.63 + 3.63i)8-s + (−0.623 − 1.08i)10-s + (1.54 − 5.75i)11-s + (−3.50 − 0.833i)13-s + (−6.11 − 2.29i)14-s − 4.51·16-s − 0.933·17-s + (−7.66 + 2.05i)19-s + (1.99 + 0.534i)20-s + (7.34 + 12.7i)22-s + 8.12i·23-s + ⋯
L(s)  = 1  + (−1.23 + 1.23i)2-s − 2.04i·4-s + (−0.0585 + 0.218i)5-s + (0.413 + 0.910i)7-s + (1.28 + 1.28i)8-s + (−0.197 − 0.341i)10-s + (0.464 − 1.73i)11-s + (−0.972 − 0.231i)13-s + (−1.63 − 0.613i)14-s − 1.12·16-s − 0.226·17-s + (−1.75 + 0.471i)19-s + (0.446 + 0.119i)20-s + (1.56 + 2.71i)22-s + 1.69i·23-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: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0950828 - 0.223471i\)
\(L(\frac12)\) \(\approx\) \(0.0950828 - 0.223471i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 2.40i)T \)
13 \( 1 + (3.50 + 0.833i)T \)
good2 \( 1 + (1.74 - 1.74i)T - 2iT^{2} \)
5 \( 1 + (0.130 - 0.488i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.54 + 5.75i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.933T + 17T^{2} \)
19 \( 1 + (7.66 - 2.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 8.12iT - 23T^{2} \)
29 \( 1 + (1.96 - 3.40i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.37 - 0.636i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-0.859 - 0.859i)T + 37iT^{2} \)
41 \( 1 + (7.84 - 2.10i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.152 - 0.0881i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.65 + 0.444i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.750 - 1.30i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.03 + 3.03i)T - 59iT^{2} \)
61 \( 1 + (6.74 + 3.89i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.37 + 1.97i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (6.54 + 1.75i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.27 + 12.2i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.64 - 8.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.66 + 1.66i)T + 83iT^{2} \)
89 \( 1 + (3.02 - 3.02i)T - 89iT^{2} \)
97 \( 1 + (0.856 - 3.19i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.62420284161948067801993529739, −9.517757316707970376368077657989, −8.848163057722444427736994202350, −8.330683184192083270147468001638, −7.50158533841563038262637379711, −6.51176619647190749426290913183, −5.83607953088945294532507765874, −5.04769444632636617805940933177, −3.28519230677645986973758730143, −1.64199692696204121278989930383, 0.17726442663493930404479677559, 1.75059084038074667296776627981, 2.52108395913616905292910112812, 4.24986496019802909547926371177, 4.56407116718366043808304388810, 6.74442730496611753825697372284, 7.30925342067724638879582929808, 8.335871724332244965064979885149, 8.986927841837363249340357866693, 9.956684113633201260506150218377

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