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Properties

Label 2-819-91.81-c1-0-26
Degree $2$
Conductor $819$
Sign $0.446 + 0.894i$
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  + (0.777 − 1.34i)2-s + (−0.208 − 0.361i)4-s + (−0.595 − 1.03i)5-s + (0.337 + 2.62i)7-s + 2.46·8-s − 1.85·10-s − 2.11·11-s + (2.86 − 2.19i)13-s + (3.79 + 1.58i)14-s + (2.33 − 4.03i)16-s + (−0.453 − 0.784i)17-s + 6.69·19-s + (−0.248 + 0.430i)20-s + (−1.64 + 2.84i)22-s + (1.79 − 3.11i)23-s + ⋯
L(s)  = 1  + (0.549 − 0.952i)2-s + (−0.104 − 0.180i)4-s + (−0.266 − 0.461i)5-s + (0.127 + 0.991i)7-s + 0.870·8-s − 0.585·10-s − 0.638·11-s + (0.793 − 0.608i)13-s + (1.01 + 0.423i)14-s + (0.582 − 1.00i)16-s + (−0.109 − 0.190i)17-s + 1.53·19-s + (−0.0555 + 0.0962i)20-s + (−0.350 + 0.607i)22-s + (0.375 − 0.649i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.446 + 0.894i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.446 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94618 - 1.20339i\)
\(L(\frac12)\) \(\approx\) \(1.94618 - 1.20339i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.337 - 2.62i)T \)
13 \( 1 + (-2.86 + 2.19i)T \)
good2 \( 1 + (-0.777 + 1.34i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.595 + 1.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 2.11T + 11T^{2} \)
17 \( 1 + (0.453 + 0.784i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.69T + 19T^{2} \)
23 \( 1 + (-1.79 + 3.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.25 - 7.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.49 - 4.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.768 - 1.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.59 + 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.41 - 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.12 + 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 8.26T + 61T^{2} \)
67 \( 1 + 3.74T + 67T^{2} \)
71 \( 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.86 + 4.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (8.87 - 15.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36732597488436673101025710555, −9.337945044687083428897283001223, −8.354808997715701027347931051218, −7.78316426228680829204825617952, −6.46519037675297529835798822532, −5.22845277925676348829411156546, −4.72749151734347562718373342415, −3.31038610949630699081647205506, −2.69251271117801454728596550229, −1.23123438728261160337940908066, 1.35950203407958789340694264690, 3.21951586875473700477922460107, 4.21802229564582279763986191557, 5.13736079566781087709963454524, 6.07367094908362610057750877159, 7.03643899714917405131761492413, 7.45024304567169091142382331585, 8.329908424752512890512217543327, 9.601828411360206450402989356346, 10.53277794909801280111062305406

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