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Properties

Label 2-819-91.45-c1-0-32
Degree $2$
Conductor $819$
Sign $0.245 + 0.969i$
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  + (0.984 − 0.984i)2-s + 0.0619i·4-s + (0.172 − 0.643i)5-s + (−2.46 − 0.960i)7-s + (2.02 + 2.02i)8-s + (−0.463 − 0.802i)10-s + (1.24 − 4.65i)11-s + (3.60 − 0.0282i)13-s + (−3.37 + 1.48i)14-s + 3.87·16-s − 0.467·17-s + (3.26 − 0.873i)19-s + (0.0398 + 0.0106i)20-s + (−3.35 − 5.81i)22-s − 6.95i·23-s + ⋯
L(s)  = 1  + (0.696 − 0.696i)2-s + 0.0309i·4-s + (0.0770 − 0.287i)5-s + (−0.931 − 0.363i)7-s + (0.717 + 0.717i)8-s + (−0.146 − 0.253i)10-s + (0.376 − 1.40i)11-s + (0.999 − 0.00782i)13-s + (−0.901 + 0.395i)14-s + 0.968·16-s − 0.113·17-s + (0.748 − 0.200i)19-s + (0.00890 + 0.00238i)20-s + (−0.715 − 1.23i)22-s − 1.45i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76782 - 1.37601i\)
\(L(\frac12)\) \(\approx\) \(1.76782 - 1.37601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.46 + 0.960i)T \)
13 \( 1 + (-3.60 + 0.0282i)T \)
good2 \( 1 + (-0.984 + 0.984i)T - 2iT^{2} \)
5 \( 1 + (-0.172 + 0.643i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.24 + 4.65i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.467T + 17T^{2} \)
19 \( 1 + (-3.26 + 0.873i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.95iT - 23T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.10 - 1.09i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.38 + 2.38i)T + 37iT^{2} \)
41 \( 1 + (3.68 - 0.986i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.42 + 1.97i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.64 - 2.58i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.20 - 3.81i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.33 - 4.33i)T - 59iT^{2} \)
61 \( 1 + (-4.21 - 2.43i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.03 + 2.42i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.19 + 0.857i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.0301 + 0.112i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.194 - 0.337i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.5 + 11.5i)T + 83iT^{2} \)
89 \( 1 + (6.83 - 6.83i)T - 89iT^{2} \)
97 \( 1 + (4.61 - 17.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.46989810415521148737154732980, −9.067056252283140772967499673843, −8.611817346516191151981773531802, −7.45607485193294720008682032777, −6.37689751401457705957106052182, −5.56349938519334178366970582226, −4.32350669044844351209599913602, −3.51217867317834432175697939981, −2.77024083090144784341358543799, −1.01523022314962138106529319060, 1.56497913605936191444562727650, 3.22278044209976580903611558761, 4.16052048347783423939754400012, 5.27520791849725827877565381802, 6.02375148542796941168230320651, 6.89792022146900342211063946657, 7.35623354141154243165151181192, 8.783864068496205572459755511475, 9.674570934077289204064609732776, 10.20791790650668808117179463594

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