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Properties

Label 2-819-91.81-c1-0-30
Degree $2$
Conductor $819$
Sign $-0.0121 + 0.999i$
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  + (1.27 − 2.20i)2-s + (−2.25 − 3.90i)4-s + (1.39 + 2.41i)5-s + (2.06 + 1.64i)7-s − 6.39·8-s + 7.10·10-s + 2.76·11-s + (2.99 + 2.01i)13-s + (6.28 − 2.46i)14-s + (−3.64 + 6.31i)16-s + (−2.94 − 5.09i)17-s + 3.40·19-s + (6.27 − 10.8i)20-s + (3.52 − 6.11i)22-s + (−3.67 + 6.36i)23-s + ⋯
L(s)  = 1  + (0.901 − 1.56i)2-s + (−1.12 − 1.95i)4-s + (0.623 + 1.07i)5-s + (0.782 + 0.623i)7-s − 2.25·8-s + 2.24·10-s + 0.834·11-s + (0.830 + 0.557i)13-s + (1.67 − 0.659i)14-s + (−0.910 + 1.57i)16-s + (−0.713 − 1.23i)17-s + 0.780·19-s + (1.40 − 2.43i)20-s + (0.752 − 1.30i)22-s + (−0.765 + 1.32i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0121 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0121 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0121 + 0.999i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0121 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.02564 - 2.05031i\)
\(L(\frac12)\) \(\approx\) \(2.02564 - 2.05031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.06 - 1.64i)T \)
13 \( 1 + (-2.99 - 2.01i)T \)
good2 \( 1 + (-1.27 + 2.20i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.39 - 2.41i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
17 \( 1 + (2.94 + 5.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.40T + 19T^{2} \)
23 \( 1 + (3.67 - 6.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.56 + 2.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.93 + 3.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.92 - 5.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.24 + 5.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.99 + 5.18i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.95 + 6.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.34 - 10.9i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 2.36i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 9.55T + 61T^{2} \)
67 \( 1 + 8.68T + 67T^{2} \)
71 \( 1 + (-1.57 + 2.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.80 - 8.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.88 + 3.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.82T + 83T^{2} \)
89 \( 1 + (-0.877 + 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.48 + 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.22436770966274696954078120283, −9.530180369735330470573557308164, −8.805737418222517198028859863274, −7.24627659083395595165195040705, −6.14414655559180364960986645657, −5.41350887762355888160879359107, −4.32613781409431516639089740804, −3.35938870403834148500644678278, −2.36530249722327871942503934275, −1.53557730294854673516762020124, 1.43600376873028875432278563480, 3.64962739174125046992789558796, 4.46724011385836866493677232079, 5.16439670230326791880719702904, 6.08950679360616622983364885881, 6.70454352659246078047243111826, 7.930775384235523259520827194244, 8.430667969988484084533498977157, 9.113953041769148646129613557152, 10.37450170810255442017961242640

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