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Properties

Label 2-819-91.47-c1-0-34
Degree $2$
Conductor $819$
Sign $-0.822 + 0.568i$
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  + (−0.788 + 0.211i)2-s + (−1.15 + 0.666i)4-s + (−0.814 − 3.03i)5-s + (1.32 − 2.28i)7-s + (1.92 − 1.92i)8-s + (1.28 + 2.22i)10-s + (−0.491 − 0.131i)11-s + (1.73 + 3.15i)13-s + (−0.562 + 2.08i)14-s + (0.221 − 0.382i)16-s + (−0.606 − 1.05i)17-s + (−0.461 − 1.72i)19-s + (2.96 + 2.96i)20-s + 0.415·22-s + (−4.51 − 2.60i)23-s + ⋯
L(s)  = 1  + (−0.557 + 0.149i)2-s + (−0.577 + 0.333i)4-s + (−0.364 − 1.35i)5-s + (0.501 − 0.865i)7-s + (0.680 − 0.680i)8-s + (0.406 + 0.703i)10-s + (−0.148 − 0.0396i)11-s + (0.481 + 0.876i)13-s + (−0.150 + 0.557i)14-s + (0.0552 − 0.0957i)16-s + (−0.147 − 0.254i)17-s + (−0.105 − 0.394i)19-s + (0.662 + 0.662i)20-s + 0.0885·22-s + (−0.940 − 0.543i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.822 + 0.568i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159922 - 0.513152i\)
\(L(\frac12)\) \(\approx\) \(0.159922 - 0.513152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.28i)T \)
13 \( 1 + (-1.73 - 3.15i)T \)
good2 \( 1 + (0.788 - 0.211i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.814 + 3.03i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.491 + 0.131i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.606 + 1.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.461 + 1.72i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.51 + 2.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.64T + 29T^{2} \)
31 \( 1 + (3.64 + 0.976i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.715 - 2.66i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (5.55 - 5.55i)T - 41iT^{2} \)
43 \( 1 + 7.46iT - 43T^{2} \)
47 \( 1 + (-4.73 + 1.26i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.30 + 7.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.648 + 2.41i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.09 + 5.25i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.91 - 7.15i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.840 + 0.840i)T + 71iT^{2} \)
73 \( 1 + (-0.632 + 2.36i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.20 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.31 - 7.31i)T - 83iT^{2} \)
89 \( 1 + (9.42 - 2.52i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.93 + 2.93i)T - 97iT^{2} \)
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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

−9.648247266212940371639237357395, −8.898986006078268805402671937711, −8.309103431061212599254740449794, −7.64204470924755411736165263145, −6.66010078318496729186724240411, −5.13508385544693061418748346316, −4.42876247016751963747555323497, −3.77687277175246603482433639522, −1.58146528317051836616774201759, −0.34033523760964306096376751106, 1.76582863622112290417176304354, 3.00588580464644739417521984989, 4.14752209684955095666642686473, 5.46719877587218539960839589333, 6.09293617015686076368519517778, 7.45387984666049328942986600318, 8.060345353123107869082888303963, 8.884870615874404147043600709556, 9.807024462163655222439147507971, 10.66683625782494070334925060166

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