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Properties

Label 2-819-91.6-c1-0-20
Degree $2$
Conductor $819$
Sign $-0.950 + 0.309i$
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  + (−0.687 − 2.56i)2-s + (−4.37 + 2.52i)4-s + (−1.17 − 1.17i)5-s + (2.40 − 1.09i)7-s + (5.73 + 5.73i)8-s + (−2.21 + 3.83i)10-s + (4.50 − 1.20i)11-s + (3.59 + 0.264i)13-s + (−4.46 − 5.42i)14-s + (5.71 − 9.90i)16-s + (3.15 + 5.46i)17-s + (1.09 − 4.08i)19-s + (8.13 + 2.17i)20-s + (−6.19 − 10.7i)22-s + (−3.67 − 2.12i)23-s + ⋯
L(s)  = 1  + (−0.486 − 1.81i)2-s + (−2.18 + 1.26i)4-s + (−0.526 − 0.526i)5-s + (0.910 − 0.414i)7-s + (2.02 + 2.02i)8-s + (−0.699 + 1.21i)10-s + (1.35 − 0.363i)11-s + (0.997 + 0.0733i)13-s + (−1.19 − 1.44i)14-s + (1.42 − 2.47i)16-s + (0.765 + 1.32i)17-s + (0.250 − 0.936i)19-s + (1.81 + 0.487i)20-s + (−1.31 − 2.28i)22-s + (−0.766 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.950 + 0.309i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.950 + 0.309i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185543 - 1.16787i\)
\(L(\frac12)\) \(\approx\) \(0.185543 - 1.16787i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.40 + 1.09i)T \)
13 \( 1 + (-3.59 - 0.264i)T \)
good2 \( 1 + (0.687 + 2.56i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (1.17 + 1.17i)T + 5iT^{2} \)
11 \( 1 + (-4.50 + 1.20i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.15 - 5.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.09 + 4.08i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.526 + 0.912i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.61 - 5.61i)T + 31iT^{2} \)
37 \( 1 + (-0.572 + 0.153i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.24 + 0.333i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.27 - 5.35i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.85 + 2.85i)T - 47iT^{2} \)
53 \( 1 - 0.398T + 53T^{2} \)
59 \( 1 + (8.26 + 2.21i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.22 - 2.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.76 + 10.3i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.31 + 1.15i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.935 + 0.935i)T - 73iT^{2} \)
79 \( 1 + 0.927T + 79T^{2} \)
83 \( 1 + (-7.79 - 7.79i)T + 83iT^{2} \)
89 \( 1 + (-1.28 - 4.78i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.65 + 6.18i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.11741277276466400303534995253, −9.016819366413305511188155753584, −8.457434594356074529622810891191, −7.933989813324673747102716061966, −6.33074120008314246112334144783, −4.73981267715350435441107599800, −4.08880902650887331028244164500, −3.32091831221213466530653767221, −1.67167694957132254090790496591, −0.926168310577783505587031456137, 1.28676448242141931990366261724, 3.65359799189750786042789115104, 4.63026564164621747617403152093, 5.67311014912537475478297006449, 6.35282749726295877675503922722, 7.37046030317487819136998664017, 7.82170670468369790205038403849, 8.690733810783331438072811373316, 9.410924564210213829553380308600, 10.23774581860428955338028780382

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