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Properties

Label 2-819-91.19-c1-0-41
Degree $2$
Conductor $819$
Sign $0.738 + 0.673i$
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  + (1.53 + 0.409i)2-s + (0.440 + 0.254i)4-s + (3.63 − 0.973i)5-s + (−1.31 − 2.29i)7-s + (−1.66 − 1.66i)8-s + 5.95·10-s + (−2.38 − 2.38i)11-s + (3.03 + 1.94i)13-s + (−1.06 − 4.05i)14-s + (−2.37 − 4.12i)16-s + (1.66 − 2.89i)17-s + (0.537 + 0.537i)19-s + (1.84 + 0.495i)20-s + (−2.67 − 4.62i)22-s + (−3.08 + 1.78i)23-s + ⋯
L(s)  = 1  + (1.08 + 0.289i)2-s + (0.220 + 0.127i)4-s + (1.62 − 0.435i)5-s + (−0.495 − 0.868i)7-s + (−0.590 − 0.590i)8-s + 1.88·10-s + (−0.718 − 0.718i)11-s + (0.842 + 0.538i)13-s + (−0.284 − 1.08i)14-s + (−0.594 − 1.03i)16-s + (0.404 − 0.701i)17-s + (0.123 + 0.123i)19-s + (0.413 + 0.110i)20-s + (−0.569 − 0.986i)22-s + (−0.643 + 0.371i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.738 + 0.673i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.738 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.75348 - 1.06689i\)
\(L(\frac12)\) \(\approx\) \(2.75348 - 1.06689i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 2.29i)T \)
13 \( 1 + (-3.03 - 1.94i)T \)
good2 \( 1 + (-1.53 - 0.409i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.63 + 0.973i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.38 + 2.38i)T + 11iT^{2} \)
17 \( 1 + (-1.66 + 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.537 - 0.537i)T + 19iT^{2} \)
23 \( 1 + (3.08 - 1.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.42 + 2.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.38 - 8.90i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.93 - 10.9i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-6.84 + 1.83i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-10.9 + 6.31i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.356 + 1.32i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.59 - 6.22i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.816 + 3.04i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 1.04iT - 61T^{2} \)
67 \( 1 + (2.09 - 2.09i)T - 67iT^{2} \)
71 \( 1 + (-7.00 - 1.87i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-2.08 - 0.559i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.54 - 9.60i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.51 + 1.51i)T + 83iT^{2} \)
89 \( 1 + (-3.05 - 0.819i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.23 + 12.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.07382498344825712258208275285, −9.417061809293461834838938259239, −8.590637540936216638622660765181, −7.17889610491067044670429640384, −6.27764589222309929099307421100, −5.70711697648179476356214477474, −4.95078155821041138833711661976, −3.83396227672360137106295286354, −2.78395872970872845190471360820, −1.10791709711099508307970368794, 2.11184269357641353891915168024, 2.69996533050021054864453193014, 3.87462261772361988085626368760, 5.19852260336548070882923713377, 5.90002267025093285557536827211, 6.20412130283303290956736073801, 7.72373865101742181021389886660, 8.895130195523219444807882446616, 9.549130599274644793307887944300, 10.41384908833149145171856155619

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