Properties

Label 2-819-91.4-c1-0-39
Degree $2$
Conductor $819$
Sign $-0.798 + 0.601i$
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  − 1.70i·2-s − 0.922·4-s + (1.84 − 1.06i)5-s + (1.43 − 2.22i)7-s − 1.84i·8-s + (−1.81 − 3.15i)10-s + (2.01 − 1.16i)11-s + (−2.64 + 2.44i)13-s + (−3.79 − 2.45i)14-s − 4.99·16-s + 0.207·17-s + (2.00 + 1.16i)19-s + (−1.70 + 0.981i)20-s + (−1.98 − 3.44i)22-s + 5.92·23-s + ⋯
L(s)  = 1  − 1.20i·2-s − 0.461·4-s + (0.824 − 0.475i)5-s + (0.543 − 0.839i)7-s − 0.651i·8-s + (−0.575 − 0.996i)10-s + (0.607 − 0.350i)11-s + (−0.733 + 0.679i)13-s + (−1.01 − 0.656i)14-s − 1.24·16-s + 0.0504·17-s + (0.460 + 0.266i)19-s + (−0.380 + 0.219i)20-s + (−0.423 − 0.734i)22-s + 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.798 + 0.601i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.798 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632563 - 1.89132i\)
\(L(\frac12)\) \(\approx\) \(0.632563 - 1.89132i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.43 + 2.22i)T \)
13 \( 1 + (2.64 - 2.44i)T \)
good2 \( 1 + 1.70iT - 2T^{2} \)
5 \( 1 + (-1.84 + 1.06i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.01 + 1.16i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.207T + 17T^{2} \)
19 \( 1 + (-2.00 - 1.16i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.92T + 23T^{2} \)
29 \( 1 + (-2.10 + 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.50 + 2.02i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.74iT - 37T^{2} \)
41 \( 1 + (-2.48 - 1.43i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.38 + 5.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (10.6 - 6.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.04 + 3.53i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.29iT - 59T^{2} \)
61 \( 1 + (0.470 - 0.815i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.52 - 2.03i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.10 - 0.638i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.60 - 2.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.77 - 9.99i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.31iT - 83T^{2} \)
89 \( 1 - 14.3iT - 89T^{2} \)
97 \( 1 + (-3.07 + 1.77i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.765284709373637862599631519150, −9.592589680524606651272299614048, −8.467490860520502879972618179070, −7.25749014251476695523898077446, −6.47120135133461900936227023667, −5.14753165514566227505852072871, −4.29036944049743720416516499782, −3.21342748395573693159317826562, −1.91331031465306938528668341603, −1.07117613470803242609312838502, 1.92827256141307301899054326874, 2.96796853575523556825359053045, 4.84726660532055847974989483417, 5.41426265050825649802146945698, 6.27740143394874543823965828393, 7.05825093022628947579755062757, 7.80211485377021891584426616996, 8.833895980878437664839912560018, 9.417975540624856580691414612234, 10.50266036359513309667883680749

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