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Properties

Label 2-819-91.24-c1-0-9
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: $\chi_{819} (388, \cdot )$
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

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

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