Properties

Label 2-819-91.4-c1-0-31
Degree $2$
Conductor $819$
Sign $0.639 + 0.768i$
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  − 0.499i·2-s + 1.75·4-s + (0.902 − 0.521i)5-s + (2.63 − 0.239i)7-s − 1.87i·8-s + (−0.260 − 0.451i)10-s + (3.43 − 1.98i)11-s + (−3.57 + 0.468i)13-s + (−0.119 − 1.31i)14-s + 2.56·16-s − 0.142·17-s + (−4.77 − 2.75i)19-s + (1.57 − 0.912i)20-s + (−0.991 − 1.71i)22-s + 4.39·23-s + ⋯
L(s)  = 1  − 0.353i·2-s + 0.875·4-s + (0.403 − 0.233i)5-s + (0.995 − 0.0904i)7-s − 0.662i·8-s + (−0.0824 − 0.142i)10-s + (1.03 − 0.598i)11-s + (−0.991 + 0.129i)13-s + (−0.0319 − 0.352i)14-s + 0.640·16-s − 0.0344·17-s + (−1.09 − 0.632i)19-s + (0.353 − 0.203i)20-s + (−0.211 − 0.366i)22-s + 0.915·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.639 + 0.768i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07142 - 0.971374i\)
\(L(\frac12)\) \(\approx\) \(2.07142 - 0.971374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.239i)T \)
13 \( 1 + (3.57 - 0.468i)T \)
good2 \( 1 + 0.499iT - 2T^{2} \)
5 \( 1 + (-0.902 + 0.521i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.43 + 1.98i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.142T + 17T^{2} \)
19 \( 1 + (4.77 + 2.75i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.39T + 23T^{2} \)
29 \( 1 + (4.19 - 7.27i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.46 - 1.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.843iT - 37T^{2} \)
41 \( 1 + (10.4 + 6.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.94 + 2.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.139 - 0.242i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.7iT - 59T^{2} \)
61 \( 1 + (-2.93 + 5.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.45 - 2.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.20 + 1.84i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.72 - 3.30i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.96 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.87iT - 83T^{2} \)
89 \( 1 + 1.74iT - 89T^{2} \)
97 \( 1 + (-2.34 + 1.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.33679718251333668209165829718, −9.226301718589092606019844242215, −8.598521807265530850298809199720, −7.34718722154842585625913104751, −6.80312045903546418282158465199, −5.68577173354688012136139347483, −4.73669179561711704648903238483, −3.52923126661019591506768901975, −2.25482742446617083838257937533, −1.30725486933704495124244846736, 1.72366327285081231684991845575, 2.48588288085043609780181726987, 4.10592857573591613857576980113, 5.12766076553305769744686596305, 6.15251282623890870959896752930, 6.85191604675138839506117281231, 7.72193331330416100161082137691, 8.445938805563638736868847314563, 9.617333402716541518737312872279, 10.34540162216779262025594233097

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