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Properties

Label 2-819-91.41-c1-0-39
Degree $2$
Conductor $819$
Sign $-0.818 + 0.573i$
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  + (1.38 + 0.369i)2-s + (0.0377 + 0.0217i)4-s + (−0.512 − 0.512i)5-s + (−2.41 − 1.09i)7-s + (−1.97 − 1.97i)8-s + (−0.517 − 0.896i)10-s + (−1.38 + 5.18i)11-s + (−0.0545 − 3.60i)13-s + (−2.92 − 2.39i)14-s + (−2.04 − 3.53i)16-s + (1.31 − 2.28i)17-s + (−5.26 + 1.41i)19-s + (−0.00817 − 0.0304i)20-s + (−3.83 + 6.64i)22-s + (−5.51 + 3.18i)23-s + ⋯
L(s)  = 1  + (0.976 + 0.261i)2-s + (0.0188 + 0.0108i)4-s + (−0.229 − 0.229i)5-s + (−0.910 − 0.412i)7-s + (−0.699 − 0.699i)8-s + (−0.163 − 0.283i)10-s + (−0.419 + 1.56i)11-s + (−0.0151 − 0.999i)13-s + (−0.781 − 0.641i)14-s + (−0.510 − 0.884i)16-s + (0.319 − 0.553i)17-s + (−1.20 + 0.323i)19-s + (−0.00182 − 0.00681i)20-s + (−0.818 + 1.41i)22-s + (−1.14 + 0.663i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.818 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.818 + 0.573i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (496, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.818 + 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185715 - 0.588634i\)
\(L(\frac12)\) \(\approx\) \(0.185715 - 0.588634i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 1.09i)T \)
13 \( 1 + (0.0545 + 3.60i)T \)
good2 \( 1 + (-1.38 - 0.369i)T + (1.73 + i)T^{2} \)
5 \( 1 + (0.512 + 0.512i)T + 5iT^{2} \)
11 \( 1 + (1.38 - 5.18i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.31 + 2.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.26 - 1.41i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.51 - 3.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.300 + 0.520i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.22 + 6.22i)T + 31iT^{2} \)
37 \( 1 + (-0.172 + 0.644i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.11 + 7.88i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-4.10 - 2.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.25 - 4.25i)T - 47iT^{2} \)
53 \( 1 + 0.282T + 53T^{2} \)
59 \( 1 + (-1.21 - 4.54i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-13.0 - 7.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 1.20i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.11 + 15.3i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.04 + 3.04i)T - 73iT^{2} \)
79 \( 1 - 4.77T + 79T^{2} \)
83 \( 1 + (2.42 + 2.42i)T + 83iT^{2} \)
89 \( 1 + (-5.75 - 1.54i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.7 - 4.21i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.918560489845393665434185416570, −9.267589909276671218321066528551, −7.923386902214368398342960480934, −7.19837502064897755068040206576, −6.18584136084923593455269923592, −5.43231914859222553150524948380, −4.35803933290905770318181487617, −3.76929039174006425589278756332, −2.42700270381155486559377218141, −0.20606960991911655533968705187, 2.32880432942548545586907323160, 3.39259747707910775930552444401, 3.98095728658650561876578562468, 5.26642639654769057763828538290, 6.06495213651208098422946560091, 6.76632960481747230491951619983, 8.302816193946697954256574769851, 8.724791843742400768447693694550, 9.746338347662108585817241251539, 10.91867192522177529821790417004

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