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Properties

Label 2-819-91.16-c1-0-15
Degree $2$
Conductor $819$
Sign $0.989 - 0.143i$
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  − 2.68·2-s + 5.20·4-s + (−0.261 + 0.452i)5-s + (0.345 + 2.62i)7-s − 8.61·8-s + (0.701 − 1.21i)10-s + (0.747 − 1.29i)11-s + (−0.735 − 3.52i)13-s + (−0.926 − 7.04i)14-s + 12.7·16-s + 1.84·17-s + (−1.75 − 3.03i)19-s + (−1.36 + 2.35i)20-s + (−2.00 + 3.47i)22-s + 7.87·23-s + ⋯
L(s)  = 1  − 1.89·2-s + 2.60·4-s + (−0.116 + 0.202i)5-s + (0.130 + 0.991i)7-s − 3.04·8-s + (0.221 − 0.384i)10-s + (0.225 − 0.390i)11-s + (−0.203 − 0.979i)13-s + (−0.247 − 1.88i)14-s + 3.17·16-s + 0.446·17-s + (−0.401 − 0.696i)19-s + (−0.304 + 0.527i)20-s + (−0.427 + 0.740i)22-s + 1.64·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.989 - 0.143i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.989 - 0.143i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.641973 + 0.0463319i\)
\(L(\frac12)\) \(\approx\) \(0.641973 + 0.0463319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.345 - 2.62i)T \)
13 \( 1 + (0.735 + 3.52i)T \)
good2 \( 1 + 2.68T + 2T^{2} \)
5 \( 1 + (0.261 - 0.452i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.747 + 1.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.84T + 17T^{2} \)
19 \( 1 + (1.75 + 3.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.87T + 23T^{2} \)
29 \( 1 + (0.977 + 1.69i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.36 + 4.09i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.37T + 37T^{2} \)
41 \( 1 + (3.34 + 5.80i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.32 + 4.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.75 - 9.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.85 - 8.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 9.87T + 59T^{2} \)
61 \( 1 + (-0.254 - 0.441i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.65 - 2.86i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.250 + 0.434i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.65 - 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.42 + 9.38i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.77T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + (0.483 - 0.837i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.09459407267167983556258088169, −9.171335398149233092370681089760, −8.803168394090574265162192907573, −7.85764944841376034278876126299, −7.18593914108332482584090671209, −6.18765646737221128322502534490, −5.29682686660611788583116904909, −3.18592275019544496205877758723, −2.35781123035376532944278463706, −0.857065293846844212762372408106, 0.897952805221499140167247468065, 1.98240748473547279324316057661, 3.47318800007181914498303153456, 4.88505422972953919783609423714, 6.53422761802695866588024246464, 6.95706251090066073076653844878, 7.83776590479793795023494569183, 8.574834472767261344872587085110, 9.392005296375252339648497580049, 10.06757500065337807671367746085

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