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Properties

Label 2-819-819.229-c1-0-22
Degree $2$
Conductor $819$
Sign $0.329 - 0.944i$
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  + (−1.60 − 1.60i)2-s + (0.794 + 1.53i)3-s + 3.14i·4-s + (2.37 + 0.637i)5-s + (1.19 − 3.74i)6-s + (−0.371 + 2.61i)7-s + (1.83 − 1.83i)8-s + (−1.73 + 2.44i)9-s + (−2.79 − 4.83i)10-s + (−0.0542 − 0.202i)11-s + (−4.84 + 2.49i)12-s + (0.522 + 3.56i)13-s + (4.79 − 3.60i)14-s + (0.909 + 4.16i)15-s + 0.394·16-s + (−1.30 − 2.26i)17-s + ⋯
L(s)  = 1  + (−1.13 − 1.13i)2-s + (0.458 + 0.888i)3-s + 1.57i·4-s + (1.06 + 0.285i)5-s + (0.487 − 1.52i)6-s + (−0.140 + 0.990i)7-s + (0.649 − 0.649i)8-s + (−0.579 + 0.815i)9-s + (−0.883 − 1.53i)10-s + (−0.0163 − 0.0610i)11-s + (−1.39 + 0.721i)12-s + (0.144 + 0.989i)13-s + (1.28 − 0.963i)14-s + (0.234 + 1.07i)15-s + 0.0986·16-s + (−0.316 − 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.801589 + 0.569321i\)
\(L(\frac12)\) \(\approx\) \(0.801589 + 0.569321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.794 - 1.53i)T \)
7 \( 1 + (0.371 - 2.61i)T \)
13 \( 1 + (-0.522 - 3.56i)T \)
good2 \( 1 + (1.60 + 1.60i)T + 2iT^{2} \)
5 \( 1 + (-2.37 - 0.637i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.0542 + 0.202i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.30 + 2.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.123 - 0.461i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.87 + 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.48 - 6.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 - 4.83i)T - 31iT^{2} \)
37 \( 1 + (2.81 + 10.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.04 + 3.88i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.49 - 3.75i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.671 - 0.671i)T + 47iT^{2} \)
53 \( 1 + (-4.32 - 7.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.61 - 1.61i)T + 59iT^{2} \)
61 \( 1 - 11.7iT - 61T^{2} \)
67 \( 1 + (3.44 + 3.44i)T + 67iT^{2} \)
71 \( 1 + (2.65 + 2.65i)T + 71iT^{2} \)
73 \( 1 + (-0.192 + 0.719i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 - 3.53T + 79T^{2} \)
83 \( 1 + (-14.7 - 3.94i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.03 + 1.34i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.2 - 4.08i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.38107459158354067343422524999, −9.432164213549927944204845930537, −9.033729752270000219197202032621, −8.613612556883265550480727672740, −7.20475518275134127352187957477, −5.92923635777288085758540182901, −4.95682239237117720571270574446, −3.48035309596785771606582533605, −2.55752600923105204639039481307, −1.84984105568021503111970323032, 0.66045422921309933036358753392, 1.79601613879451649641516649955, 3.40708622021100633284458931274, 5.20381871416544459853641158181, 6.19775171578250601075008429795, 6.68416854865027579092417779997, 7.65221669773638782016072275249, 8.194625744631948174177233932951, 9.028154119177125322389595341889, 9.859384632155500688507594368914

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