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Properties

Label 2-819-91.4-c1-0-9
Degree $2$
Conductor $819$
Sign $0.445 + 0.895i$
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  − 2.62i·2-s − 4.90·4-s + (−2.98 + 1.72i)5-s + (−2.63 − 0.240i)7-s + 7.61i·8-s + (4.53 + 7.85i)10-s + (4.24 − 2.45i)11-s + (−3.59 + 0.286i)13-s + (−0.632 + 6.92i)14-s + 10.2·16-s + 2.16·17-s + (1.83 + 1.05i)19-s + (14.6 − 8.45i)20-s + (−6.44 − 11.1i)22-s + 7.75·23-s + ⋯
L(s)  = 1  − 1.85i·2-s − 2.45·4-s + (−1.33 + 0.772i)5-s + (−0.995 − 0.0910i)7-s + 2.69i·8-s + (1.43 + 2.48i)10-s + (1.28 − 0.739i)11-s + (−0.996 + 0.0793i)13-s + (−0.169 + 1.84i)14-s + 2.55·16-s + 0.525·17-s + (0.420 + 0.242i)19-s + (3.27 − 1.89i)20-s + (−1.37 − 2.37i)22-s + 1.61·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.445 + 0.895i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604700 - 0.374716i\)
\(L(\frac12)\) \(\approx\) \(0.604700 - 0.374716i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.240i)T \)
13 \( 1 + (3.59 - 0.286i)T \)
good2 \( 1 + 2.62iT - 2T^{2} \)
5 \( 1 + (2.98 - 1.72i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.24 + 2.45i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 2.16T + 17T^{2} \)
19 \( 1 + (-1.83 - 1.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.75T + 23T^{2} \)
29 \( 1 + (4.61 - 7.99i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.456iT - 37T^{2} \)
41 \( 1 + (-3.36 - 1.94i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.634 + 1.09i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.62 + 3.24i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.681 + 1.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 4.35iT - 59T^{2} \)
61 \( 1 + (7.21 - 12.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.34 + 3.66i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.96 + 4.02i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.12 - 2.38i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.74 - 8.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.37iT - 83T^{2} \)
89 \( 1 - 2.15iT - 89T^{2} \)
97 \( 1 + (12.9 - 7.48i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.37481772804845505310220466362, −9.342045046587484820059293137347, −8.944428907977468131915584556767, −7.60187444381916072883106928197, −6.79161899793456185935271832105, −5.26775803434449238883077798251, −3.92679059134823510445695014645, −3.50779498295073579234298424998, −2.74176240689522284846023235205, −0.925439902615489067858913968156, 0.51315467088227690805426257183, 3.52058761706748297040113522734, 4.37110511453881838514310382631, 5.11110434026470183054901901136, 6.19947389814707663351877721114, 7.26609558618965019107769524505, 7.39783512272106412521422018127, 8.524657477224105801446085250057, 9.347867586203613238894536883189, 9.652137789374542844611361069005

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