Properties

Label 2-819-91.45-c1-0-6
Degree $2$
Conductor $819$
Sign $-0.319 + 0.947i$
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  + (−1.09 + 1.09i)2-s − 0.419i·4-s + (−0.745 + 2.78i)5-s + (−1.80 + 1.93i)7-s + (−1.73 − 1.73i)8-s + (−2.24 − 3.88i)10-s + (−1.41 + 5.28i)11-s + (−0.662 + 3.54i)13-s + (−0.136 − 4.11i)14-s + 4.66·16-s + 4.36·17-s + (−1.39 + 0.373i)19-s + (1.16 + 0.312i)20-s + (−4.25 − 7.37i)22-s − 8.37i·23-s + ⋯
L(s)  = 1  + (−0.777 + 0.777i)2-s − 0.209i·4-s + (−0.333 + 1.24i)5-s + (−0.683 + 0.730i)7-s + (−0.614 − 0.614i)8-s + (−0.708 − 1.22i)10-s + (−0.427 + 1.59i)11-s + (−0.183 + 0.982i)13-s + (−0.0363 − 1.09i)14-s + 1.16·16-s + 1.05·17-s + (−0.320 + 0.0857i)19-s + (0.261 + 0.0699i)20-s + (−0.907 − 1.57i)22-s − 1.74i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.319 + 0.947i$
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: \((2,\ 819,\ (\ :1/2),\ -0.319 + 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.299241 - 0.416773i\)
\(L(\frac12)\) \(\approx\) \(0.299241 - 0.416773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.93i)T \)
13 \( 1 + (0.662 - 3.54i)T \)
good2 \( 1 + (1.09 - 1.09i)T - 2iT^{2} \)
5 \( 1 + (0.745 - 2.78i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.41 - 5.28i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 4.36T + 17T^{2} \)
19 \( 1 + (1.39 - 0.373i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 8.37iT - 23T^{2} \)
29 \( 1 + (0.882 - 1.52i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.770 + 0.206i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-3.86 - 3.86i)T + 37iT^{2} \)
41 \( 1 + (3.88 - 1.03i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.58 - 3.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-8.52 - 2.28i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.139 - 0.241i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.16 + 5.16i)T - 59iT^{2} \)
61 \( 1 + (-4.10 - 2.37i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 0.502i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (11.3 + 3.04i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.72 + 13.9i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.431 + 0.746i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.29 + 4.29i)T + 83iT^{2} \)
89 \( 1 + (4.21 - 4.21i)T - 89iT^{2} \)
97 \( 1 + (0.575 - 2.14i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.36540795794709167117547316414, −9.940244764267188556828867653508, −9.097571692330590352524520479135, −8.162897889768058123245354665228, −7.26380494935390384898963645889, −6.78431356176967534389423343324, −6.07857583224812320983100058177, −4.59048526866989112442941946841, −3.30077832337028925266655119355, −2.34924493034560229958288149471, 0.37609454661940130378672527261, 1.15337120083687006694609779832, 2.96317516615394254661817117295, 3.80074338569382222156799447961, 5.40151171250275618596434314541, 5.77714923943105048885780656781, 7.44864617253973080571159423322, 8.262331113152177430643493094872, 8.835696130898863613559956646031, 9.793347914896619701806943000539

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