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Properties

Label 2-819-91.41-c1-0-9
Degree $2$
Conductor $819$
Sign $0.985 + 0.172i$
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  + (−2.37 − 0.636i)2-s + (3.50 + 2.02i)4-s + (0.498 + 0.498i)5-s + (−2.62 − 0.304i)7-s + (−3.56 − 3.56i)8-s + (−0.866 − 1.50i)10-s + (−0.184 + 0.688i)11-s + (−3.17 + 1.70i)13-s + (6.05 + 2.39i)14-s + (2.15 + 3.73i)16-s + (2.27 − 3.93i)17-s + (3.24 − 0.870i)19-s + (0.739 + 2.75i)20-s + (0.876 − 1.51i)22-s + (1.67 − 0.964i)23-s + ⋯
L(s)  = 1  + (−1.68 − 0.450i)2-s + (1.75 + 1.01i)4-s + (0.222 + 0.222i)5-s + (−0.993 − 0.114i)7-s + (−1.26 − 1.26i)8-s + (−0.274 − 0.474i)10-s + (−0.0556 + 0.207i)11-s + (−0.881 + 0.471i)13-s + (1.61 + 0.640i)14-s + (0.538 + 0.932i)16-s + (0.551 − 0.954i)17-s + (0.745 − 0.199i)19-s + (0.165 + 0.616i)20-s + (0.186 − 0.323i)22-s + (0.348 − 0.201i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.985 + 0.172i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.985 + 0.172i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569643 - 0.0494688i\)
\(L(\frac12)\) \(\approx\) \(0.569643 - 0.0494688i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 + 0.304i)T \)
13 \( 1 + (3.17 - 1.70i)T \)
good2 \( 1 + (2.37 + 0.636i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.498 - 0.498i)T + 5iT^{2} \)
11 \( 1 + (0.184 - 0.688i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.27 + 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.24 + 0.870i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.67 + 0.964i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.185 + 0.322i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.53 - 3.53i)T + 31iT^{2} \)
37 \( 1 + (0.545 - 2.03i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.11 - 11.6i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.38 - 3.68i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.55 + 3.55i)T - 47iT^{2} \)
53 \( 1 - 4.97T + 53T^{2} \)
59 \( 1 + (-1.03 - 3.85i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-10.0 - 5.81i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.5 + 3.37i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.10 + 7.86i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.608 + 0.608i)T - 73iT^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 + (-2.25 - 2.25i)T + 83iT^{2} \)
89 \( 1 + (-17.5 - 4.70i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-8.73 + 2.34i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.990033590488191675836954298644, −9.541683301619321167581054688154, −8.816556495876871753196796007866, −7.69470290237946763436311483230, −7.08769454402851575895812501817, −6.30825312475011111312048586384, −4.81812013326498491671345688603, −3.13682397299642941694065045434, −2.42398303522582079708055719387, −0.819703057830718746825499853116, 0.73530953186242604821900311908, 2.23023207339111775809716692651, 3.55867648794578623353993014848, 5.41899922314916882538429418917, 6.09339700976921552796981602281, 7.18450859516869164787278194824, 7.68104055565532267952837955078, 8.762761108920426521036251848653, 9.317300256948974625811281146285, 10.10507959172893057923066899128

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