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Properties

Label 2-819-91.74-c1-0-2
Degree $2$
Conductor $819$
Sign $-0.713 - 0.700i$
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  − 1.06·2-s − 0.864·4-s + (−1.19 − 2.06i)5-s + (−0.813 + 2.51i)7-s + 3.05·8-s + (1.26 + 2.19i)10-s + (−0.333 − 0.577i)11-s + (2.19 − 2.85i)13-s + (0.866 − 2.68i)14-s − 1.52·16-s − 1.41·17-s + (1.78 − 3.08i)19-s + (1.02 + 1.78i)20-s + (0.355 + 0.615i)22-s − 5.98·23-s + ⋯
L(s)  = 1  − 0.753·2-s − 0.432·4-s + (−0.532 − 0.921i)5-s + (−0.307 + 0.951i)7-s + 1.07·8-s + (0.401 + 0.694i)10-s + (−0.100 − 0.174i)11-s + (0.609 − 0.792i)13-s + (0.231 − 0.716i)14-s − 0.380·16-s − 0.343·17-s + (0.408 − 0.707i)19-s + (0.230 + 0.398i)20-s + (0.0757 + 0.131i)22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.713 - 0.700i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (802, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0565849 + 0.138328i\)
\(L(\frac12)\) \(\approx\) \(0.0565849 + 0.138328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.813 - 2.51i)T \)
13 \( 1 + (-2.19 + 2.85i)T \)
good2 \( 1 + 1.06T + 2T^{2} \)
5 \( 1 + (1.19 + 2.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.333 + 0.577i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.98T + 23T^{2} \)
29 \( 1 + (0.647 - 1.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.89T + 37T^{2} \)
41 \( 1 + (5.26 - 9.11i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.22 - 9.04i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.54 - 9.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.39 - 5.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 5.15T + 59T^{2} \)
61 \( 1 + (2.41 - 4.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.78 + 4.82i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.01 + 10.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.05 - 7.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.00 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.44T + 83T^{2} \)
89 \( 1 + 1.82T + 89T^{2} \)
97 \( 1 + (7.88 + 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.38346968552379652835848713209, −9.449333931891985121693666539830, −8.790474337310817332805901120538, −8.318933136298925016279853385971, −7.51111838475315190270482942112, −6.13278923394352720870942516387, −5.17773549423081063265214181375, −4.35180428998075773360593086883, −3.06596718797754246523700901791, −1.34337261139963004001069335050, 0.10539871373488336726684088014, 1.85750899770044225538748438301, 3.73831109420659514026949937289, 4.02955425762015684462938007678, 5.56104580106498382257472271066, 6.89957511006669871736755111922, 7.31427662981799005175695759418, 8.235576293006685211315225003835, 9.080003662918128991091552655157, 10.08196071037816934982341815568

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