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Properties

Label 2-819-91.47-c1-0-41
Degree $2$
Conductor $819$
Sign $-0.0629 + 0.998i$
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  + (2.60 − 0.697i)2-s + (4.55 − 2.63i)4-s + (−0.654 − 2.44i)5-s + (−2.54 − 0.722i)7-s + (6.22 − 6.22i)8-s + (−3.40 − 5.89i)10-s + (2.08 + 0.557i)11-s + (−1.44 + 3.30i)13-s + (−7.13 − 0.104i)14-s + (6.59 − 11.4i)16-s + (−0.700 − 1.21i)17-s + (0.541 + 2.02i)19-s + (−9.40 − 9.40i)20-s + 5.80·22-s + (−1.13 − 0.657i)23-s + ⋯
L(s)  = 1  + (1.84 − 0.493i)2-s + (2.27 − 1.31i)4-s + (−0.292 − 1.09i)5-s + (−0.962 − 0.272i)7-s + (2.19 − 2.19i)8-s + (−1.07 − 1.86i)10-s + (0.627 + 0.168i)11-s + (−0.401 + 0.915i)13-s + (−1.90 − 0.0279i)14-s + (1.64 − 2.85i)16-s + (−0.169 − 0.294i)17-s + (0.124 + 0.463i)19-s + (−2.10 − 2.10i)20-s + 1.23·22-s + (−0.237 − 0.137i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0629 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0629 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0629 + 0.998i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0629 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84160 - 3.02649i\)
\(L(\frac12)\) \(\approx\) \(2.84160 - 3.02649i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.722i)T \)
13 \( 1 + (1.44 - 3.30i)T \)
good2 \( 1 + (-2.60 + 0.697i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.654 + 2.44i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.08 - 0.557i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.700 + 1.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.541 - 2.02i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.13 + 0.657i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + (-7.03 - 1.88i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.591 - 2.20i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.69 - 2.69i)T - 41iT^{2} \)
43 \( 1 - 0.437iT - 43T^{2} \)
47 \( 1 + (7.74 - 2.07i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.26 + 2.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.02 + 7.54i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.57 - 3.79i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.146 + 0.548i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-10.7 - 10.7i)T + 71iT^{2} \)
73 \( 1 + (-3.18 + 11.8i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.19 - 12.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.82 - 3.82i)T - 83iT^{2} \)
89 \( 1 + (-0.0501 + 0.0134i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (9.43 - 9.43i)T - 97iT^{2} \)
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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

−10.10931433345685525830312053576, −9.522822505501116033511054944613, −8.238254385599710676885952941327, −6.79229236198748939902294059547, −6.47464847674074976146114003319, −5.21474970721338927255201866270, −4.48461178897807595724908386391, −3.81499747590620717161301927816, −2.68034221118504623083592526929, −1.26759466712247758131471150008, 2.56965827303437054876518078693, 3.18237158804177424382980398558, 4.00895413517692619248338728608, 5.17103373811800441620262296632, 6.17010222304465261404976519556, 6.65830211042150233789943244242, 7.37619462182785863350894022338, 8.394001517652391012185216060284, 9.899724868696102645036692458420, 10.74685855190057641535107265983

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