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Properties

Label 2-819-91.19-c1-0-30
Degree $2$
Conductor $819$
Sign $-0.421 + 0.906i$
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  + (−1.07 − 0.286i)2-s + (−0.667 − 0.385i)4-s + (3.71 − 0.994i)5-s + (−1.35 + 2.27i)7-s + (2.17 + 2.17i)8-s − 4.26·10-s + (−4.06 − 4.06i)11-s + (−2.56 − 2.53i)13-s + (2.10 − 2.04i)14-s + (−0.931 − 1.61i)16-s + (2.31 − 4.00i)17-s + (1.10 + 1.10i)19-s + (−2.86 − 0.766i)20-s + (3.18 + 5.51i)22-s + (−3.73 + 2.15i)23-s + ⋯
L(s)  = 1  + (−0.757 − 0.202i)2-s + (−0.333 − 0.192i)4-s + (1.66 − 0.444i)5-s + (−0.512 + 0.858i)7-s + (0.767 + 0.767i)8-s − 1.34·10-s + (−1.22 − 1.22i)11-s + (−0.711 − 0.702i)13-s + (0.562 − 0.546i)14-s + (−0.232 − 0.403i)16-s + (0.560 − 0.970i)17-s + (0.252 + 0.252i)19-s + (−0.640 − 0.171i)20-s + (0.679 + 1.17i)22-s + (−0.778 + 0.449i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.421 + 0.906i$
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: \((2,\ 819,\ (\ :1/2),\ -0.421 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.449916 - 0.705368i\)
\(L(\frac12)\) \(\approx\) \(0.449916 - 0.705368i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 2.27i)T \)
13 \( 1 + (2.56 + 2.53i)T \)
good2 \( 1 + (1.07 + 0.286i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-3.71 + 0.994i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.06 + 4.06i)T + 11iT^{2} \)
17 \( 1 + (-2.31 + 4.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 1.10i)T + 19iT^{2} \)
23 \( 1 + (3.73 - 2.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.38 + 4.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.993 + 3.70i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.653 - 2.44i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.40 + 1.18i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.11 + 7.89i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.36 - 2.37i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.64 + 13.6i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.55iT - 61T^{2} \)
67 \( 1 + (4.37 - 4.37i)T - 67iT^{2} \)
71 \( 1 + (-7.06 - 1.89i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.11 - 0.298i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.33 + 7.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.07 + 1.07i)T + 83iT^{2} \)
89 \( 1 + (8.76 + 2.34i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-1.63 + 6.11i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.874912649715286835096311471746, −9.369726464438657176672978220634, −8.452694780400484256459906634693, −7.79907411061718821149734351485, −6.15894969795639237866072008392, −5.47630934880764983114384558964, −5.08340257237216311457423243914, −2.94885659354917444118612057026, −2.10864260584548138881438637671, −0.53163994196002546457277482167, 1.57300935112077384203276316828, 2.76933302931592846962208099972, 4.27029252066508580824512224564, 5.22222275436431321941956781177, 6.43491966449968047305629127378, 7.13867991844221706705006787083, 7.86630339366278816087408999598, 9.082267309160966341662245697593, 9.787289206706755742277015144695, 10.23636828577694752949038026061

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