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Properties

Label 2-819-91.24-c1-0-6
Degree $2$
Conductor $819$
Sign $-0.470 - 0.882i$
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  + (−1.64 + 0.441i)2-s + (0.784 − 0.453i)4-s + (−1.35 − 0.363i)5-s + (0.501 + 2.59i)7-s + (1.31 − 1.31i)8-s + 2.39·10-s + (−1.07 + 1.07i)11-s + (1.37 − 3.33i)13-s + (−1.97 − 4.05i)14-s + (−2.49 + 4.32i)16-s + (2.58 + 4.47i)17-s + (3.42 − 3.42i)19-s + (−1.23 + 0.329i)20-s + (1.29 − 2.23i)22-s + (−1.86 − 1.07i)23-s + ⋯
L(s)  = 1  + (−1.16 + 0.311i)2-s + (0.392 − 0.226i)4-s + (−0.607 − 0.162i)5-s + (0.189 + 0.981i)7-s + (0.466 − 0.466i)8-s + 0.758·10-s + (−0.323 + 0.323i)11-s + (0.380 − 0.924i)13-s + (−0.527 − 1.08i)14-s + (−0.623 + 1.08i)16-s + (0.626 + 1.08i)17-s + (0.786 − 0.786i)19-s + (−0.275 + 0.0737i)20-s + (0.275 − 0.477i)22-s + (−0.388 − 0.224i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.470 - 0.882i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.470 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.272868 + 0.454521i\)
\(L(\frac12)\) \(\approx\) \(0.272868 + 0.454521i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.501 - 2.59i)T \)
13 \( 1 + (-1.37 + 3.33i)T \)
good2 \( 1 + (1.64 - 0.441i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.35 + 0.363i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.07 - 1.07i)T - 11iT^{2} \)
17 \( 1 + (-2.58 - 4.47i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.42 + 3.42i)T - 19iT^{2} \)
23 \( 1 + (1.86 + 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.744 - 1.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.506 - 1.89i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.45 - 9.16i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.34 - 1.70i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (7.27 + 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.66 - 6.20i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.87 - 3.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.19 - 8.17i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.23iT - 61T^{2} \)
67 \( 1 + (-9.21 - 9.21i)T + 67iT^{2} \)
71 \( 1 + (-2.71 + 0.726i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (14.2 - 3.82i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.67 - 6.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0684 + 0.0684i)T - 83iT^{2} \)
89 \( 1 + (10.7 - 2.88i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.39 - 12.6i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.23964161472386168659673599045, −9.606684883864909003513165152935, −8.552203216791271773402897189119, −8.187291040104827777925256778724, −7.48550493378909059715723129867, −6.33210336533320947445649708877, −5.33335016852759866149861334303, −4.18742056512278551835612380307, −2.88254425636839650841282370790, −1.25234864471684670365226122431, 0.45092143903181107289101705071, 1.75884670471141682925639050432, 3.37409827080591950053331148242, 4.37104684470833576306908757047, 5.51703881993370529116920614340, 6.92701203422336915666836882337, 7.73665683657091673022625595818, 8.099737697945445731760947542596, 9.364754210777866561367196843183, 9.793390350560618926187853887691

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