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Properties

Label 2-819-91.16-c1-0-35
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

\(L(1)\) \(\approx\) \(4.86860 + 0.351372i\)
\(L(\frac12)\) \(\approx\) \(4.86860 + 0.351372i\)
\(L(\frac{3}{2})\) 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

−10.63754779247333399618230862104, −9.575030847525437154460511385881, −8.304765925109245156913069706355, −7.39931433379095550851246899203, −6.36188355776400665330672865014, −5.61161471112106925252774724247, −4.97373030377229333618379784346, −4.01319203647339934973391094256, −2.81344049562769101693686250979, −2.05033768684713423116976262529, 1.80646665491294722243739245364, 2.94353776446375115178601872856, 4.21474245241739368922791718482, 4.40739431725660477843991065497, 5.84425074416129219486027438849, 6.40317628592939633759230206585, 7.29104382525771227370147900577, 8.102997183065885609772155150145, 9.679260053794156292431073730174, 10.80761796072649441515615007791

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