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Properties

Label 2-819-91.59-c1-0-17
Degree $2$
Conductor $819$
Sign $0.989 + 0.141i$
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  + (−0.111 − 0.111i)2-s − 1.97i·4-s + (2.13 + 0.570i)5-s + (−0.399 + 2.61i)7-s + (−0.442 + 0.442i)8-s + (−0.173 − 0.300i)10-s + (5.42 + 1.45i)11-s + (−1.34 − 3.34i)13-s + (0.335 − 0.246i)14-s − 3.85·16-s + 3.59·17-s + (1.24 + 4.65i)19-s + (1.12 − 4.20i)20-s + (−0.441 − 0.764i)22-s + 4.45i·23-s + ⋯
L(s)  = 1  + (−0.0786 − 0.0786i)2-s − 0.987i·4-s + (0.952 + 0.255i)5-s + (−0.151 + 0.988i)7-s + (−0.156 + 0.156i)8-s + (−0.0548 − 0.0950i)10-s + (1.63 + 0.438i)11-s + (−0.371 − 0.928i)13-s + (0.0896 − 0.0658i)14-s − 0.963·16-s + 0.870·17-s + (0.286 + 1.06i)19-s + (0.252 − 0.941i)20-s + (−0.0941 − 0.163i)22-s + 0.928i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.989 + 0.141i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (514, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.989 + 0.141i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.85628 - 0.131955i\)
\(L(\frac12)\) \(\approx\) \(1.85628 - 0.131955i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.399 - 2.61i)T \)
13 \( 1 + (1.34 + 3.34i)T \)
good2 \( 1 + (0.111 + 0.111i)T + 2iT^{2} \)
5 \( 1 + (-2.13 - 0.570i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-5.42 - 1.45i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 + (-1.24 - 4.65i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 4.45iT - 23T^{2} \)
29 \( 1 + (1.02 - 1.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.643 - 2.40i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-7.22 + 7.22i)T - 37iT^{2} \)
41 \( 1 + (1.34 + 5.01i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-4.12 + 2.38i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.22 + 4.56i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.201 - 0.348i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 1.93i)T + 59iT^{2} \)
61 \( 1 + (-1.61 - 0.930i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.78 + 6.65i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.63 + 13.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.45 - 1.19i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.45 + 4.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.3 - 10.3i)T - 83iT^{2} \)
89 \( 1 + (-12.7 - 12.7i)T + 89iT^{2} \)
97 \( 1 + (3.52 + 0.944i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.949593060034789015893303010039, −9.573277929607345672828805481015, −8.897430115376868844574691880369, −7.58414167706672945171931361898, −6.44575274218394853517860600914, −5.77261524270794179985332455276, −5.27930263019474296273259400435, −3.70324541085394488493305746686, −2.31254527737250680128385301909, −1.36053024932815741622741746227, 1.19326832848248473482068050586, 2.72932432204111464619362499015, 3.93012348635870152776212786389, 4.62277074321567803339839680483, 6.17670282403144163745799916279, 6.76405303950615042741239304310, 7.62382451463778289201654440843, 8.682846529930758361076180929478, 9.433627587238906811922986786174, 9.935537005954454331366578591751

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