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Properties

Label 2-819-117.41-c1-0-42
Degree $2$
Conductor $819$
Sign $7.70e-5 - 0.999i$
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  + (−1.06 + 0.285i)2-s + (0.611 + 1.62i)3-s + (−0.677 + 0.391i)4-s + (4.22 − 1.13i)5-s + (−1.11 − 1.55i)6-s + (0.707 + 0.707i)7-s + (2.17 − 2.17i)8-s + (−2.25 + 1.98i)9-s + (−4.18 + 2.41i)10-s + (−0.432 − 1.61i)11-s + (−1.04 − 0.859i)12-s + (−2.06 + 2.95i)13-s + (−0.955 − 0.551i)14-s + (4.41 + 6.15i)15-s + (−0.910 + 1.57i)16-s + (−0.720 + 1.24i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.201i)2-s + (0.352 + 0.935i)3-s + (−0.338 + 0.195i)4-s + (1.89 − 0.506i)5-s + (−0.454 − 0.633i)6-s + (0.267 + 0.267i)7-s + (0.767 − 0.767i)8-s + (−0.750 + 0.660i)9-s + (−1.32 + 0.763i)10-s + (−0.130 − 0.486i)11-s + (−0.302 − 0.248i)12-s + (−0.571 + 0.820i)13-s + (−0.255 − 0.147i)14-s + (1.14 + 1.59i)15-s + (−0.227 + 0.394i)16-s + (−0.174 + 0.302i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (7.70e-5 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (7.70e-5 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $7.70e-5 - 0.999i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 7.70e-5 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02022 + 1.02015i\)
\(L(\frac12)\) \(\approx\) \(1.02022 + 1.02015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.611 - 1.62i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (2.06 - 2.95i)T \)
good2 \( 1 + (1.06 - 0.285i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-4.22 + 1.13i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.432 + 1.61i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.720 - 1.24i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.10 - 4.11i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 + (0.483 + 0.278i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.393 + 1.46i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.95 - 7.30i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.28 + 2.28i)T + 41iT^{2} \)
43 \( 1 + 6.80iT - 43T^{2} \)
47 \( 1 + (-9.28 - 2.48i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 13.5iT - 53T^{2} \)
59 \( 1 + (-2.71 - 0.727i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 - 8.52T + 61T^{2} \)
67 \( 1 + (1.15 - 1.15i)T - 67iT^{2} \)
71 \( 1 + (13.2 - 3.55i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (5.23 + 5.23i)T + 73iT^{2} \)
79 \( 1 + (2.40 + 4.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.06 + 3.96i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.58 + 1.22i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.60 - 8.60i)T - 97iT^{2} \)
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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

−10.18373401108473510344374722609, −9.409092208365469702856551401061, −8.954305811057150871626755958182, −8.422790792274684525027753584007, −7.13977463350010468919087186226, −5.88579933896112867730311649719, −5.14858397632245007210374713529, −4.27696626870025237262474949085, −2.78558784404552648141355328769, −1.52184989034376871586110226885, 0.989085260514085523731756557831, 2.08683707146917313169486743557, 2.85405048803790366004294752530, 5.01777716410934072243560385890, 5.59152143425681664052025025818, 6.85669009111746061361570103788, 7.34902479372710785230805884341, 8.579019890242253502877323045787, 9.253742558098165048175979021259, 9.860271987721298202553892340171

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