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Properties

Label 2-819-819.86-c1-0-55
Degree $2$
Conductor $819$
Sign $0.974 + 0.225i$
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  + (−1.84 + 1.84i)2-s + (0.853 + 1.50i)3-s − 4.82i·4-s + (−0.833 − 0.223i)5-s + (−4.36 − 1.20i)6-s + (−2.31 + 1.27i)7-s + (5.22 + 5.22i)8-s + (−1.54 + 2.57i)9-s + (1.95 − 1.12i)10-s + (−0.360 + 0.0965i)11-s + (7.27 − 4.12i)12-s + (−1.36 − 3.33i)13-s + (1.92 − 6.64i)14-s + (−0.375 − 1.44i)15-s − 9.65·16-s + (−0.674 − 1.16i)17-s + ⋯
L(s)  = 1  + (−1.30 + 1.30i)2-s + (0.492 + 0.870i)3-s − 2.41i·4-s + (−0.372 − 0.0999i)5-s + (−1.78 − 0.492i)6-s + (−0.875 + 0.482i)7-s + (1.84 + 1.84i)8-s + (−0.514 + 0.857i)9-s + (0.617 − 0.356i)10-s + (−0.108 + 0.0291i)11-s + (2.10 − 1.18i)12-s + (−0.378 − 0.925i)13-s + (0.513 − 1.77i)14-s + (−0.0968 − 0.373i)15-s − 2.41·16-s + (−0.163 − 0.283i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.974 + 0.225i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.974 + 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226741 - 0.0258810i\)
\(L(\frac12)\) \(\approx\) \(0.226741 - 0.0258810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.853 - 1.50i)T \)
7 \( 1 + (2.31 - 1.27i)T \)
13 \( 1 + (1.36 + 3.33i)T \)
good2 \( 1 + (1.84 - 1.84i)T - 2iT^{2} \)
5 \( 1 + (0.833 + 0.223i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.360 - 0.0965i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.674 + 1.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.22 - 0.597i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.50 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.36 + 3.09i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.05 + 4.05i)T + 31iT^{2} \)
37 \( 1 + (2.15 + 8.02i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.598 + 2.23i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.36 + 3.67i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.509 - 0.509i)T + 47iT^{2} \)
53 \( 1 + (-8.86 + 5.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.09 + 7.09i)T + 59iT^{2} \)
61 \( 1 - 0.717T + 61T^{2} \)
67 \( 1 + (-6.50 - 6.50i)T + 67iT^{2} \)
71 \( 1 + (8.47 - 8.47i)T - 71iT^{2} \)
73 \( 1 + (-8.34 - 2.23i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + 9.28T + 79T^{2} \)
83 \( 1 + (-2.97 - 0.796i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (10.4 - 2.79i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.86 + 10.7i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.860136663021117775832576154723, −9.244184708439478958823522390424, −8.561260286333390406281966033704, −7.79439503142739416596877950925, −7.06659891529295793704771407929, −5.82430055132548106018696750301, −5.31449299202073077664165196693, −3.89864997340883904381769125318, −2.45431977046362237599957256907, −0.17598992982618809239857476679, 1.22638957298472183971783685140, 2.47600557886421435427175054189, 3.26351225057040903451816468930, 4.28706284381727940769350029292, 6.51410472123279333594198855003, 7.10684674589791958287613894968, 7.956928671142962623486205259345, 8.831769193599067410662504006127, 9.286267451882022535509506797821, 10.30550660437108351378674725330

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