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Properties

Label 2-819-91.41-c1-0-40
Degree $2$
Conductor $819$
Sign $-0.104 + 0.994i$
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  + (2.11 + 0.567i)2-s + (2.43 + 1.40i)4-s + (−3.00 − 3.00i)5-s + (−2.49 + 0.883i)7-s + (1.26 + 1.26i)8-s + (−4.65 − 8.06i)10-s + (0.698 − 2.60i)11-s + (−0.373 − 3.58i)13-s + (−5.78 + 0.455i)14-s + (−0.857 − 1.48i)16-s + (0.599 − 1.03i)17-s + (−1.89 + 0.507i)19-s + (−3.09 − 11.5i)20-s + (2.95 − 5.12i)22-s + (4.65 − 2.68i)23-s + ⋯
L(s)  = 1  + (1.49 + 0.401i)2-s + (1.21 + 0.703i)4-s + (−1.34 − 1.34i)5-s + (−0.942 + 0.333i)7-s + (0.445 + 0.445i)8-s + (−1.47 − 2.55i)10-s + (0.210 − 0.785i)11-s + (−0.103 − 0.994i)13-s + (−1.54 + 0.121i)14-s + (−0.214 − 0.371i)16-s + (0.145 − 0.251i)17-s + (−0.434 + 0.116i)19-s + (−0.691 − 2.57i)20-s + (0.630 − 1.09i)22-s + (0.970 − 0.560i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.104 + 0.994i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.104 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22628 - 1.36245i\)
\(L(\frac12)\) \(\approx\) \(1.22628 - 1.36245i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 - 0.883i)T \)
13 \( 1 + (0.373 + 3.58i)T \)
good2 \( 1 + (-2.11 - 0.567i)T + (1.73 + i)T^{2} \)
5 \( 1 + (3.00 + 3.00i)T + 5iT^{2} \)
11 \( 1 + (-0.698 + 2.60i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.599 + 1.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.89 - 0.507i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.65 + 2.68i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.47 + 2.56i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.36 - 3.36i)T + 31iT^{2} \)
37 \( 1 + (1.03 - 3.87i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.42 - 5.32i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.78 + 5.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.97 - 2.97i)T - 47iT^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 + (3.14 + 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.55 - 2.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.15 - 1.11i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.800 + 2.98i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.78 - 4.78i)T - 73iT^{2} \)
79 \( 1 + 1.16T + 79T^{2} \)
83 \( 1 + (3.24 + 3.24i)T + 83iT^{2} \)
89 \( 1 + (-4.80 - 1.28i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.7 + 3.95i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.02741623075873333421170973013, −8.838072852646366247469168586216, −8.287404192034648721795104155276, −7.22020156997364418386000989521, −6.29080006738005974751653669487, −5.32193488646801508272917880715, −4.70727732523021410786429451449, −3.64546765098961240824507627298, −3.07120684887627616469288025752, −0.53730168073482270296955938178, 2.33144436390676016444679445430, 3.37119086390443006053961876869, 3.89126892217498751614407426438, 4.72079442286396671690786846321, 6.16971113731944974569905463786, 6.90652352987255370684608535266, 7.33727114513920403969000626304, 8.776403283136943702911896484149, 10.02827877158391512756190316343, 10.75493037181040996237240552166

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