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Properties

Label 2-819-91.47-c1-0-35
Degree $2$
Conductor $819$
Sign $-0.813 + 0.581i$
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  + (0.746 − 0.200i)2-s + (−1.21 + 0.701i)4-s + (−0.472 − 1.76i)5-s + (−0.210 + 2.63i)7-s + (−1.85 + 1.85i)8-s + (−0.705 − 1.22i)10-s + (−0.990 − 0.265i)11-s + (−0.266 − 3.59i)13-s + (0.370 + 2.01i)14-s + (0.386 − 0.669i)16-s + (−2.60 − 4.50i)17-s + (−1.36 − 5.07i)19-s + (1.81 + 1.81i)20-s − 0.792·22-s + (−0.730 − 0.421i)23-s + ⋯
L(s)  = 1  + (0.527 − 0.141i)2-s + (−0.607 + 0.350i)4-s + (−0.211 − 0.788i)5-s + (−0.0796 + 0.996i)7-s + (−0.657 + 0.657i)8-s + (−0.222 − 0.386i)10-s + (−0.298 − 0.0800i)11-s + (−0.0738 − 0.997i)13-s + (0.0989 + 0.537i)14-s + (0.0966 − 0.167i)16-s + (−0.630 − 1.09i)17-s + (−0.312 − 1.16i)19-s + (0.404 + 0.404i)20-s − 0.169·22-s + (−0.152 − 0.0879i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.813 + 0.581i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167153 - 0.521277i\)
\(L(\frac12)\) \(\approx\) \(0.167153 - 0.521277i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.210 - 2.63i)T \)
13 \( 1 + (0.266 + 3.59i)T \)
good2 \( 1 + (-0.746 + 0.200i)T + (1.73 - i)T^{2} \)
5 \( 1 + (0.472 + 1.76i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.990 + 0.265i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (2.60 + 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.36 + 5.07i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.730 + 0.421i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + (5.69 + 1.52i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.61 - 6.03i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.0927 + 0.0927i)T - 41iT^{2} \)
43 \( 1 - 7.36iT - 43T^{2} \)
47 \( 1 + (2.17 - 0.583i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.38 + 5.86i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.60 - 9.73i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.13 - 0.653i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.11 + 4.16i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.02 - 6.02i)T + 71iT^{2} \)
73 \( 1 + (-2.93 + 10.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.16 + 8.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.16 + 4.16i)T - 83iT^{2} \)
89 \( 1 + (-7.49 + 2.00i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.49 + 2.49i)T - 97iT^{2} \)
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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

−9.546564912302215595131296778949, −9.065629463865549666507072354142, −8.347434325113456648667839096322, −7.50146862889363040771181835788, −6.08291418039354327071774708181, −5.11733012755250613402060678588, −4.72997382810142740003151112991, −3.36635072541938937310139297515, −2.43102983668972126425379874153, −0.22135178741819384126818531762, 1.83707682720384697180060087536, 3.78408637791628949733105942522, 3.89264003161989769933022511236, 5.23865450639143641645846205249, 6.24495962090339600423575946215, 6.97925269501303910220900988813, 7.85987383538685457632165116139, 9.010847018892927085277903468557, 9.780875023416156729162459214461, 10.71204912669503292084859063296

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