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Properties

Label 2-819-91.23-c1-0-3
Degree $2$
Conductor $819$
Sign $0.183 + 0.983i$
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  + 2.69i·2-s − 5.27·4-s + (0.933 + 0.539i)5-s + (2.30 + 1.29i)7-s − 8.84i·8-s + (−1.45 + 2.51i)10-s + (−4.50 − 2.59i)11-s + (−1.40 + 3.31i)13-s + (−3.49 + 6.22i)14-s + 13.3·16-s − 5.38·17-s + (−5.71 + 3.29i)19-s + (−4.92 − 2.84i)20-s + (7.01 − 12.1i)22-s − 2.00·23-s + ⋯
L(s)  = 1  + 1.90i·2-s − 2.63·4-s + (0.417 + 0.241i)5-s + (0.871 + 0.489i)7-s − 3.12i·8-s + (−0.459 + 0.796i)10-s + (−1.35 − 0.783i)11-s + (−0.390 + 0.920i)13-s + (−0.934 + 1.66i)14-s + 3.32·16-s − 1.30·17-s + (−1.31 + 0.756i)19-s + (−1.10 − 0.636i)20-s + (1.49 − 2.58i)22-s − 0.417·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.183 + 0.983i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.183 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391923 - 0.325498i\)
\(L(\frac12)\) \(\approx\) \(0.391923 - 0.325498i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.30 - 1.29i)T \)
13 \( 1 + (1.40 - 3.31i)T \)
good2 \( 1 - 2.69iT - 2T^{2} \)
5 \( 1 + (-0.933 - 0.539i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.50 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 5.38T + 17T^{2} \)
19 \( 1 + (5.71 - 3.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.00T + 23T^{2} \)
29 \( 1 + (2.56 + 4.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.21 + 2.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.82iT - 37T^{2} \)
41 \( 1 + (1.70 - 0.982i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.82 - 6.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.33 - 4.81i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.752 - 1.30i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.61iT - 59T^{2} \)
61 \( 1 + (2.74 + 4.74i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.83 - 2.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.55 + 4.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.88 + 2.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.65 + 2.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.99iT - 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + (-6.64 - 3.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.61894645109203674977929698916, −9.748035028438141652816360480848, −8.720259133869169278828246515654, −8.237532717552370550727117994758, −7.55270836288930491354849047075, −6.28256571804838357744257656676, −6.04185938005477516527162937028, −4.86137942827907416708208624296, −4.27977768300231798693867778034, −2.31788776049574549957025709447, 0.23837924860044547193057200494, 1.91927643136043130674567508123, 2.48852541205357780923246494011, 3.93488004830565596438613667401, 4.83797083532028941002649097295, 5.36243476518223059026048035683, 7.25211236777851395195486320349, 8.298953888026312962545839531706, 8.950840855266986329775185170024, 9.962302530622844160475364139692

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