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Properties

Label 2-819-117.50-c1-0-1
Degree $2$
Conductor $819$
Sign $0.988 + 0.152i$
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  + (−0.563 − 2.10i)2-s + (−1.71 + 0.246i)3-s + (−2.37 + 1.37i)4-s + (−0.775 − 2.89i)5-s + (1.48 + 3.46i)6-s + (−0.707 + 0.707i)7-s + (1.14 + 1.14i)8-s + (2.87 − 0.845i)9-s + (−5.64 + 3.26i)10-s + (−1.60 + 0.430i)11-s + (3.73 − 2.93i)12-s + (−2.85 + 2.20i)13-s + (1.88 + 1.08i)14-s + (2.04 + 4.76i)15-s + (−0.984 + 1.70i)16-s + (0.438 − 0.759i)17-s + ⋯
L(s)  = 1  + (−0.398 − 1.48i)2-s + (−0.989 + 0.142i)3-s + (−1.18 + 0.685i)4-s + (−0.346 − 1.29i)5-s + (0.606 + 1.41i)6-s + (−0.267 + 0.267i)7-s + (0.403 + 0.403i)8-s + (0.959 − 0.281i)9-s + (−1.78 + 1.03i)10-s + (−0.484 + 0.129i)11-s + (1.07 − 0.847i)12-s + (−0.792 + 0.610i)13-s + (0.503 + 0.290i)14-s + (0.527 + 1.23i)15-s + (−0.246 + 0.426i)16-s + (0.106 − 0.184i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.988 + 0.152i$
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: \((2,\ 819,\ (\ :1/2),\ 0.988 + 0.152i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236985 - 0.0182206i\)
\(L(\frac12)\) \(\approx\) \(0.236985 - 0.0182206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 - 0.246i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (2.85 - 2.20i)T \)
good2 \( 1 + (0.563 + 2.10i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (0.775 + 2.89i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.60 - 0.430i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.438 + 0.759i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.110 + 0.0295i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 4.42T + 23T^{2} \)
29 \( 1 + (-5.46 - 3.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.55 - 1.75i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.53 - 2.02i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.17 + 2.17i)T - 41iT^{2} \)
43 \( 1 + 0.0653iT - 43T^{2} \)
47 \( 1 + (-2.10 + 7.86i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 + (1.26 - 4.72i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 - 6.24T + 61T^{2} \)
67 \( 1 + (3.88 + 3.88i)T + 67iT^{2} \)
71 \( 1 + (0.779 + 2.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.741 + 0.741i)T - 73iT^{2} \)
79 \( 1 + (-8.18 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.52 - 2.55i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.63 - 9.85i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (8.02 + 8.02i)T + 97iT^{2} \)
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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

−10.28053992784598694227969321905, −9.571069274599084161887669189902, −8.971818404426145003091502369463, −7.938028755877137581001118578620, −6.71455908838758144214721539404, −5.44407437909174046783143431120, −4.65279729892968879289749150420, −3.83326267259726262267885665301, −2.29326514202530552882724939417, −1.02382750359174636531629645875, 0.18676673020873654009773882627, 2.70242650608117452652701729796, 4.22732477996205280663304206353, 5.38827850002742178855877814345, 6.14800403554347702978635945743, 6.76785366240615998053642496158, 7.64870357146506402364969045003, 7.88755659413856247453821994461, 9.481045132394066916939282424745, 10.20805749914417924485911209609

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