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Properties

Label 2-819-117.2-c1-0-18
Degree $2$
Conductor $819$
Sign $-0.560 - 0.828i$
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  + (1.24 + 1.24i)2-s + (−0.599 − 1.62i)3-s + 1.11i·4-s + (−0.587 + 2.19i)5-s + (1.28 − 2.77i)6-s + (−0.258 + 0.965i)7-s + (1.09 − 1.09i)8-s + (−2.28 + 1.94i)9-s + (−3.46 + 2.00i)10-s + (−2.29 + 2.29i)11-s + (1.81 − 0.670i)12-s + (−2.31 + 2.76i)13-s + (−1.52 + 0.883i)14-s + (3.91 − 0.358i)15-s + 4.98·16-s + (−0.515 + 0.893i)17-s + ⋯
L(s)  = 1  + (0.883 + 0.883i)2-s + (−0.345 − 0.938i)3-s + 0.559i·4-s + (−0.262 + 0.979i)5-s + (0.523 − 1.13i)6-s + (−0.0978 + 0.365i)7-s + (0.388 − 0.388i)8-s + (−0.760 + 0.649i)9-s + (−1.09 + 0.633i)10-s + (−0.692 + 0.692i)11-s + (0.525 − 0.193i)12-s + (−0.641 + 0.767i)13-s + (−0.408 + 0.236i)14-s + (1.01 − 0.0926i)15-s + 1.24·16-s + (−0.125 + 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.560 - 0.828i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.560 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752717 + 1.41820i\)
\(L(\frac12)\) \(\approx\) \(0.752717 + 1.41820i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.599 + 1.62i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (2.31 - 2.76i)T \)
good2 \( 1 + (-1.24 - 1.24i)T + 2iT^{2} \)
5 \( 1 + (0.587 - 2.19i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.29 - 2.29i)T - 11iT^{2} \)
17 \( 1 + (0.515 - 0.893i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.09 - 4.07i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.536 - 0.928i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.61iT - 29T^{2} \)
31 \( 1 + (-0.634 - 0.169i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.27 - 8.47i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.51 - 1.47i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.91 + 5.14i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.876 + 3.27i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + 4.95iT - 53T^{2} \)
59 \( 1 + (7.44 - 7.44i)T - 59iT^{2} \)
61 \( 1 + (1.16 + 2.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.01 - 11.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.76 + 1.81i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.26 + 5.26i)T + 73iT^{2} \)
79 \( 1 + (-3.20 + 5.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.5 - 3.89i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (7.37 + 1.97i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-13.6 - 3.65i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.55648534200714028078576695967, −9.872377840773772760993703264380, −8.374308788017959947351345597691, −7.42978122043333415046973345892, −7.03820594659042168971195017865, −6.24393714810544374584429289499, −5.44804490998796809886097061694, −4.50977690674832717013115252123, −3.14441550771236088812403372005, −1.90150786124238418717232965283, 0.60872606347997991478316162220, 2.66684272398498094833752227484, 3.52166096654464357231970176735, 4.59732485117112650417201432566, 5.01485306900767014095864357865, 5.87349080271484324914884060961, 7.48992257157217398919822784721, 8.422994521322239878966467910501, 9.274938745644502374671242331725, 10.27870836908085963594530886604

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