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Properties

Label 2-819-91.6-c1-0-9
Degree $2$
Conductor $819$
Sign $-0.752 - 0.658i$
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  + (0.500 + 1.86i)2-s + (−1.50 + 0.870i)4-s + (−2.44 − 2.44i)5-s + (2.02 + 1.70i)7-s + (0.353 + 0.353i)8-s + (3.33 − 5.78i)10-s + (−2.85 + 0.764i)11-s + (3.60 − 0.0697i)13-s + (−2.16 + 4.63i)14-s + (−2.22 + 3.85i)16-s + (0.667 + 1.15i)17-s + (−1.32 + 4.94i)19-s + (5.80 + 1.55i)20-s + (−2.85 − 4.94i)22-s + (7.61 + 4.39i)23-s + ⋯
L(s)  = 1  + (0.354 + 1.32i)2-s + (−0.754 + 0.435i)4-s + (−1.09 − 1.09i)5-s + (0.765 + 0.643i)7-s + (0.124 + 0.124i)8-s + (1.05 − 1.82i)10-s + (−0.860 + 0.230i)11-s + (0.999 − 0.0193i)13-s + (−0.579 + 1.23i)14-s + (−0.556 + 0.963i)16-s + (0.161 + 0.280i)17-s + (−0.304 + 1.13i)19-s + (1.29 + 0.347i)20-s + (−0.609 − 1.05i)22-s + (1.58 + 0.916i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.752 - 0.658i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.551912 + 1.47012i\)
\(L(\frac12)\) \(\approx\) \(0.551912 + 1.47012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 - 1.70i)T \)
13 \( 1 + (-3.60 + 0.0697i)T \)
good2 \( 1 + (-0.500 - 1.86i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (2.44 + 2.44i)T + 5iT^{2} \)
11 \( 1 + (2.85 - 0.764i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.667 - 1.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.32 - 4.94i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-7.61 - 4.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.97 - 6.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.04 + 2.04i)T + 31iT^{2} \)
37 \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.45 + 0.390i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.212 - 0.122i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.13 + 1.13i)T - 47iT^{2} \)
53 \( 1 + 2.62T + 53T^{2} \)
59 \( 1 + (3.96 + 1.06i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (7.82 - 4.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.97 + 11.1i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-11.0 - 2.96i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.06 + 1.06i)T - 73iT^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 + (-10.1 - 10.1i)T + 83iT^{2} \)
89 \( 1 + (4.11 + 15.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.28 + 12.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.85171464999035667476513688775, −9.169236375095730049302417553652, −8.537199692986549233597136594223, −7.87566239039080808203880773362, −7.40009768636586195449214803588, −6.00380725832830159175913670042, −5.28361005123082393208784125327, −4.65857549393106873222783595438, −3.59407605921668743858173664769, −1.56843233393864301374327774964, 0.74910043858862778267398301129, 2.45865056970595867845335279258, 3.26782235742716937063176559122, 4.14096397462073819659931173163, 4.95705678773476445803941654272, 6.59281703518119301334884642751, 7.42027698806711859028923182679, 8.103674785508158553137537793785, 9.292238911266259800412656795510, 10.58801371613796545270281162492

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