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Properties

Label 2-819-91.34-c1-0-42
Degree $2$
Conductor $819$
Sign $-0.0404 + 0.999i$
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  + (0.403 + 0.403i)2-s − 1.67i·4-s + (1.03 − 1.03i)5-s + (−0.450 − 2.60i)7-s + (1.48 − 1.48i)8-s + 0.832·10-s + (0.596 − 0.596i)11-s + (3.59 − 0.296i)13-s + (0.869 − 1.23i)14-s − 2.15·16-s − 7.34·17-s + (−3.59 + 3.59i)19-s + (−1.72 − 1.72i)20-s + 0.481·22-s − 4.44i·23-s + ⋯
L(s)  = 1  + (0.284 + 0.284i)2-s − 0.837i·4-s + (0.461 − 0.461i)5-s + (−0.170 − 0.985i)7-s + (0.523 − 0.523i)8-s + 0.263·10-s + (0.179 − 0.179i)11-s + (0.996 − 0.0822i)13-s + (0.232 − 0.329i)14-s − 0.539·16-s − 1.78·17-s + (−0.824 + 0.824i)19-s + (−0.386 − 0.386i)20-s + 0.102·22-s − 0.926i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0404 + 0.999i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.0404 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22664 - 1.27732i\)
\(L(\frac12)\) \(\approx\) \(1.22664 - 1.27732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.450 + 2.60i)T \)
13 \( 1 + (-3.59 + 0.296i)T \)
good2 \( 1 + (-0.403 - 0.403i)T + 2iT^{2} \)
5 \( 1 + (-1.03 + 1.03i)T - 5iT^{2} \)
11 \( 1 + (-0.596 + 0.596i)T - 11iT^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 + (3.59 - 3.59i)T - 19iT^{2} \)
23 \( 1 + 4.44iT - 23T^{2} \)
29 \( 1 - 3.54T + 29T^{2} \)
31 \( 1 + (-1.27 + 1.27i)T - 31iT^{2} \)
37 \( 1 + (-2.88 + 2.88i)T - 37iT^{2} \)
41 \( 1 + (-1.23 + 1.23i)T - 41iT^{2} \)
43 \( 1 + 8.66iT - 43T^{2} \)
47 \( 1 + (-2.52 - 2.52i)T + 47iT^{2} \)
53 \( 1 - 9.79T + 53T^{2} \)
59 \( 1 + (1.08 + 1.08i)T + 59iT^{2} \)
61 \( 1 - 7.10iT - 61T^{2} \)
67 \( 1 + (-8.76 - 8.76i)T + 67iT^{2} \)
71 \( 1 + (-1.46 - 1.46i)T + 71iT^{2} \)
73 \( 1 + (-0.103 - 0.103i)T + 73iT^{2} \)
79 \( 1 + 4.79T + 79T^{2} \)
83 \( 1 + (-12.3 + 12.3i)T - 83iT^{2} \)
89 \( 1 + (6.89 + 6.89i)T + 89iT^{2} \)
97 \( 1 + (-6.05 + 6.05i)T - 97iT^{2} \)
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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

−10.23499531950199865996588850137, −9.102688517122325599283655162391, −8.511838056106179804478297983745, −7.14353877102906315840908538979, −6.41261948774662201278489397623, −5.75354517072603257125561434803, −4.54444906090058345174936024300, −3.94512128294279010809082144159, −2.06396097415801544588797427015, −0.801199029837687554194185906222, 2.10336824675040336954284860000, 2.81013640333195791431162422317, 4.01630274708668366078168004969, 4.94919939676390653963026701599, 6.32240791716874737926869958180, 6.71416477710478074417508893999, 8.101938362468790234199919273062, 8.764308942401454821811156414353, 9.449912118581306677634800094241, 10.73052360858660239225222456702

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