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Properties

Label 2-819-117.50-c1-0-0
Degree $2$
Conductor $819$
Sign $0.884 + 0.467i$
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  + (−0.663 − 2.47i)2-s + (−0.692 − 1.58i)3-s + (−3.95 + 2.28i)4-s + (−0.816 − 3.04i)5-s + (−3.47 + 2.76i)6-s + (−0.707 + 0.707i)7-s + (4.66 + 4.66i)8-s + (−2.03 + 2.19i)9-s + (−7.00 + 4.04i)10-s + (0.561 − 0.150i)11-s + (6.37 + 4.70i)12-s + (3.52 + 0.764i)13-s + (2.22 + 1.28i)14-s + (−4.27 + 3.40i)15-s + (3.87 − 6.71i)16-s + (−2.47 + 4.28i)17-s + ⋯
L(s)  = 1  + (−0.469 − 1.75i)2-s + (−0.400 − 0.916i)3-s + (−1.97 + 1.14i)4-s + (−0.365 − 1.36i)5-s + (−1.41 + 1.13i)6-s + (−0.267 + 0.267i)7-s + (1.64 + 1.64i)8-s + (−0.679 + 0.733i)9-s + (−2.21 + 1.27i)10-s + (0.169 − 0.0453i)11-s + (1.83 + 1.35i)12-s + (0.977 + 0.211i)13-s + (0.593 + 0.342i)14-s + (−1.10 + 0.879i)15-s + (0.969 − 1.67i)16-s + (−0.599 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.884 + 0.467i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.884 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102941 - 0.0255346i\)
\(L(\frac12)\) \(\approx\) \(0.102941 - 0.0255346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.692 + 1.58i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-3.52 - 0.764i)T \)
good2 \( 1 + (0.663 + 2.47i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.816 + 3.04i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.561 + 0.150i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.47 - 4.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.25 - 1.67i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 6.69T + 23T^{2} \)
29 \( 1 + (1.72 + 0.993i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.40 - 1.71i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (8.31 + 2.22i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.75 - 1.75i)T - 41iT^{2} \)
43 \( 1 + 1.53iT - 43T^{2} \)
47 \( 1 + (1.97 - 7.37i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 13.7iT - 53T^{2} \)
59 \( 1 + (0.728 - 2.71i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 3.69T + 61T^{2} \)
67 \( 1 + (0.933 + 0.933i)T + 67iT^{2} \)
71 \( 1 + (-0.126 - 0.473i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 73iT^{2} \)
79 \( 1 + (-4.07 - 7.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (13.8 + 3.71i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.71 - 6.39i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \)
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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

−10.51112258656694411885616235307, −9.178295145551440187050645599289, −8.598701698475232721358117232178, −8.279272638858971184516547583047, −6.76402192559495778325832780354, −5.54787473192460214427296638977, −4.45689091441019780624875486590, −3.53923393201526157066774313307, −2.01547530582373550407120880363, −1.25945563175077308482052926161, 0.07216155120302275678138193829, 3.19200771754312500864078843596, 4.22260877139923096850211247406, 5.21888738828401219008057570390, 6.24174660536307140361220348199, 6.80060221110895830173792353997, 7.41382156444065655717914528513, 8.768193644500049848478578849479, 9.071968677801394249188309157009, 10.27543610106355498208011826625

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