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Properties

Label 2-819-117.2-c1-0-20
Degree $2$
Conductor $819$
Sign $0.695 - 0.718i$
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  + (−0.849 − 0.849i)2-s + (1.47 + 0.911i)3-s − 0.555i·4-s + (−0.107 + 0.402i)5-s + (−0.477 − 2.02i)6-s + (0.258 − 0.965i)7-s + (−2.17 + 2.17i)8-s + (1.33 + 2.68i)9-s + (0.434 − 0.250i)10-s + (−2.92 + 2.92i)11-s + (0.506 − 0.817i)12-s + (−0.880 + 3.49i)13-s + (−1.04 + 0.600i)14-s + (−0.526 + 0.494i)15-s + 2.58·16-s + (1.41 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.600 − 0.600i)2-s + (0.850 + 0.526i)3-s − 0.277i·4-s + (−0.0482 + 0.180i)5-s + (−0.194 − 0.827i)6-s + (0.0978 − 0.365i)7-s + (−0.767 + 0.767i)8-s + (0.446 + 0.894i)9-s + (0.137 − 0.0792i)10-s + (−0.881 + 0.881i)11-s + (0.146 − 0.236i)12-s + (−0.244 + 0.969i)13-s + (−0.278 + 0.160i)14-s + (−0.135 + 0.127i)15-s + 0.645·16-s + (0.343 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12262 + 0.475410i\)
\(L(\frac12)\) \(\approx\) \(1.12262 + 0.475410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.47 - 0.911i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (0.880 - 3.49i)T \)
good2 \( 1 + (0.849 + 0.849i)T + 2iT^{2} \)
5 \( 1 + (0.107 - 0.402i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.92 - 2.92i)T - 11iT^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.98 - 7.40i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.07 - 3.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.85iT - 29T^{2} \)
31 \( 1 + (0.978 + 0.262i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.93 + 10.9i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (1.56 - 0.418i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.46 + 3.72i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.213 + 0.795i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 9.18iT - 53T^{2} \)
59 \( 1 + (9.86 - 9.86i)T - 59iT^{2} \)
61 \( 1 + (4.35 + 7.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.04 - 7.64i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.20 + 1.12i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (3.54 + 3.54i)T + 73iT^{2} \)
79 \( 1 + (-4.65 + 8.05i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.71 - 0.728i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.44 - 1.99i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.5 - 3.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.30327354358167443399444581146, −9.495074611171762441846102076364, −9.082328860205280246678465342410, −7.77690826824991589044814183818, −7.40389459878365094418989698519, −5.80468891498808917890700640321, −4.84220726065473126062550129350, −3.75498571094398504921711245570, −2.55477311877574022008263699127, −1.63986369193898012263858201905, 0.66819708492760238928372115267, 2.70080787382015150678103748194, 3.24601457490398951718888255033, 4.81265715916782075248992599750, 6.07494787879384785284257379564, 6.89493117820638623958117014915, 7.909776790485919445025587665666, 8.246528878646877381967996077753, 8.933376141544317616426900628644, 9.791999589126062571051736018523

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