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Properties

Label 2-819-7.2-c1-0-6
Degree $2$
Conductor $819$
Sign $-0.811 + 0.583i$
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  + (−1.08 + 1.87i)2-s + (−1.34 − 2.33i)4-s + (−0.758 + 1.31i)5-s + (1.40 + 2.24i)7-s + 1.51·8-s + (−1.64 − 2.84i)10-s + (2.04 + 3.54i)11-s − 13-s + (−5.73 + 0.208i)14-s + (1.05 − 1.82i)16-s + (2.48 + 4.29i)17-s + (1.19 − 2.07i)19-s + 4.09·20-s − 8.87·22-s + (−1.28 + 2.23i)23-s + ⋯
L(s)  = 1  + (−0.766 + 1.32i)2-s + (−0.674 − 1.16i)4-s + (−0.339 + 0.587i)5-s + (0.531 + 0.847i)7-s + 0.536·8-s + (−0.519 − 0.900i)10-s + (0.617 + 1.06i)11-s − 0.277·13-s + (−1.53 + 0.0558i)14-s + (0.263 − 0.457i)16-s + (0.601 + 1.04i)17-s + (0.275 − 0.476i)19-s + 0.915·20-s − 1.89·22-s + (−0.268 + 0.465i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.811 + 0.583i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (352, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.811 + 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242362 - 0.752203i\)
\(L(\frac12)\) \(\approx\) \(0.242362 - 0.752203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.40 - 2.24i)T \)
13 \( 1 + T \)
good2 \( 1 + (1.08 - 1.87i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (0.758 - 1.31i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.48 - 4.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 2.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.28 - 2.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 + (-0.182 - 0.316i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.96 + 3.39i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.34T + 41T^{2} \)
43 \( 1 + 8.47T + 43T^{2} \)
47 \( 1 + (-2.16 + 3.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.08 + 8.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.50 + 9.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.83 - 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.44 - 9.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.18T + 71T^{2} \)
73 \( 1 + (2.58 + 4.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.47 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (-1.87 + 3.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−10.45484237743963118672867437381, −9.542841780667818587179941041785, −8.961101805039957791268109692954, −7.983154238727736162014244855274, −7.41686936029645408271463845273, −6.63157149228354777281799745510, −5.73568517054149149519000986377, −4.85929904131032808346299642667, −3.42832496978347353650243201080, −1.81723509901332654573551577768, 0.54116257578783712287751903744, 1.49056538207339677389803185395, 3.02003423736657260186756874123, 3.90646597278875259110835963972, 4.92954147562423869787209198217, 6.25455277370097968501399159156, 7.61746907557387241084124659996, 8.234173699034294064787847631094, 9.092084607678206712749825660903, 9.783603010237384320619974203234

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