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Properties

Label 2-819-91.76-c1-0-34
Degree $2$
Conductor $819$
Sign $-0.907 + 0.420i$
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  + (0.607 − 2.26i)2-s + (−3.03 − 1.75i)4-s + (1.12 − 1.12i)5-s + (1.68 + 2.04i)7-s + (−2.50 + 2.50i)8-s + (−1.87 − 3.23i)10-s + (3.03 + 0.813i)11-s + (1.04 − 3.44i)13-s + (5.64 − 2.57i)14-s + (0.649 + 1.12i)16-s + (0.320 − 0.555i)17-s + (−2.04 − 7.61i)19-s + (−5.40 + 1.44i)20-s + (3.68 − 6.38i)22-s + (0.126 − 0.0730i)23-s + ⋯
L(s)  = 1  + (0.429 − 1.60i)2-s + (−1.51 − 0.877i)4-s + (0.503 − 0.503i)5-s + (0.636 + 0.771i)7-s + (−0.886 + 0.886i)8-s + (−0.591 − 1.02i)10-s + (0.914 + 0.245i)11-s + (0.290 − 0.956i)13-s + (1.51 − 0.689i)14-s + (0.162 + 0.281i)16-s + (0.0778 − 0.134i)17-s + (−0.468 − 1.74i)19-s + (−1.20 + 0.323i)20-s + (0.786 − 1.36i)22-s + (0.0263 − 0.0152i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.907 + 0.420i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.907 + 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.463542 - 2.10492i\)
\(L(\frac12)\) \(\approx\) \(0.463542 - 2.10492i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 - 2.04i)T \)
13 \( 1 + (-1.04 + 3.44i)T \)
good2 \( 1 + (-0.607 + 2.26i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-1.12 + 1.12i)T - 5iT^{2} \)
11 \( 1 + (-3.03 - 0.813i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.320 + 0.555i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 + 7.61i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.126 + 0.0730i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.49 + 2.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.73 - 4.73i)T - 31iT^{2} \)
37 \( 1 + (-3.75 - 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-5.60 - 1.50i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.42 - 1.40i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.22 + 2.22i)T + 47iT^{2} \)
53 \( 1 - 7.32T + 53T^{2} \)
59 \( 1 + (4.00 - 1.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.90 - 2.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.366 - 1.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (13.8 - 3.70i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.99 + 4.99i)T + 73iT^{2} \)
79 \( 1 + 0.632T + 79T^{2} \)
83 \( 1 + (-1.07 + 1.07i)T - 83iT^{2} \)
89 \( 1 + (-3.51 + 13.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.0487 + 0.181i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.01382116262601429838387650052, −9.081009842419668666750453658037, −8.818597208524340065453949275086, −7.35584403528665199994820548050, −5.93841625394912524387979906293, −5.08349131146849594259789294068, −4.34207274890243087778657687828, −3.08664532044997524051841181205, −2.11825083568908323209797342036, −1.05252403126850127535246019769, 1.75813655744188431566555859477, 3.84881100282574981435809504875, 4.31970183293796525442772133616, 5.68350729598459379697554934298, 6.24620645324083607871847849347, 7.03648960387592753284540036544, 7.78827946367572640025347929627, 8.615802200033173346712788346143, 9.499000004134538486888545115654, 10.54940403506280618779642947955

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