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Properties

Label 2-819-91.80-c1-0-27
Degree $2$
Conductor $819$
Sign $0.863 - 0.504i$
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  + (0.720 + 2.68i)2-s + (−4.97 + 2.87i)4-s + (0.470 − 1.75i)5-s + (1.29 − 2.30i)7-s + (−7.38 − 7.38i)8-s + 5.05·10-s + (−3.13 − 3.13i)11-s + (2.12 − 2.91i)13-s + (7.13 + 1.81i)14-s + (8.78 − 15.2i)16-s + (−0.0321 − 0.0557i)17-s + (2.15 + 2.15i)19-s + (2.70 + 10.0i)20-s + (6.17 − 10.6i)22-s + (−2.78 − 1.60i)23-s + ⋯
L(s)  = 1  + (0.509 + 1.90i)2-s + (−2.48 + 1.43i)4-s + (0.210 − 0.784i)5-s + (0.489 − 0.872i)7-s + (−2.61 − 2.61i)8-s + 1.59·10-s + (−0.945 − 0.945i)11-s + (0.590 − 0.807i)13-s + (1.90 + 0.486i)14-s + (2.19 − 3.80i)16-s + (−0.00780 − 0.0135i)17-s + (0.494 + 0.494i)19-s + (0.604 + 2.25i)20-s + (1.31 − 2.27i)22-s + (−0.580 − 0.335i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.863 - 0.504i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34709 + 0.364665i\)
\(L(\frac12)\) \(\approx\) \(1.34709 + 0.364665i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 2.30i)T \)
13 \( 1 + (-2.12 + 2.91i)T \)
good2 \( 1 + (-0.720 - 2.68i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.470 + 1.75i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.13 + 3.13i)T + 11iT^{2} \)
17 \( 1 + (0.0321 + 0.0557i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.15 - 2.15i)T + 19iT^{2} \)
23 \( 1 + (2.78 + 1.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.41 + 4.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.02 - 1.34i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-4.21 + 1.12i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.12 + 7.93i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.22 + 1.86i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.06 + 1.08i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.30 - 5.71i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.49 - 1.20i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.19iT - 61T^{2} \)
67 \( 1 + (-4.45 + 4.45i)T - 67iT^{2} \)
71 \( 1 + (-1.98 - 7.40i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.61 - 6.03i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.639 - 1.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.90 - 8.90i)T + 83iT^{2} \)
89 \( 1 + (0.376 + 1.40i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.79 - 2.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.01671607917111599472096853764, −9.007746457793874590253935774345, −8.127500755532249878373246859922, −7.892015503496431049857504426495, −6.84746325992396594619778752127, −5.63891150845330573098809023469, −5.41301570999509130306320299993, −4.28954735937536326310136824446, −3.41694863316237953937026708438, −0.60664428545998872341913998446, 1.70672398813871944514554151870, 2.46454121307305441593159928881, 3.38678345295506025978518111218, 4.61252621295736614727914816740, 5.28754348970518059925025298706, 6.35110880440181632936603425123, 7.87423273227291918654682025171, 8.961681338206865853759477289590, 9.625885024731528646089977511251, 10.35110210813013801171689116049

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