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Properties

Label 2-819-91.54-c1-0-11
Degree $2$
Conductor $819$
Sign $0.452 - 0.891i$
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  + (0.926 − 0.926i)2-s + 0.284i·4-s + (−0.409 + 0.109i)5-s + (−2.25 + 1.38i)7-s + (2.11 + 2.11i)8-s + (−0.277 + 0.481i)10-s + (−1.35 + 0.362i)11-s + (−3.54 + 0.648i)13-s + (−0.809 + 3.36i)14-s + 3.35·16-s + 6.94·17-s + (−0.143 + 0.535i)19-s + (−0.0312 − 0.116i)20-s + (−0.918 + 1.59i)22-s + 7.84i·23-s + ⋯
L(s)  = 1  + (0.654 − 0.654i)2-s + 0.142i·4-s + (−0.183 + 0.0491i)5-s + (−0.852 + 0.522i)7-s + (0.748 + 0.748i)8-s + (−0.0879 + 0.152i)10-s + (−0.408 + 0.109i)11-s + (−0.983 + 0.179i)13-s + (−0.216 + 0.900i)14-s + 0.837·16-s + 1.68·17-s + (−0.0328 + 0.122i)19-s + (−0.00698 − 0.0260i)20-s + (−0.195 + 0.339i)22-s + 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.452 - 0.891i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36851 + 0.840592i\)
\(L(\frac12)\) \(\approx\) \(1.36851 + 0.840592i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 - 1.38i)T \)
13 \( 1 + (3.54 - 0.648i)T \)
good2 \( 1 + (-0.926 + 0.926i)T - 2iT^{2} \)
5 \( 1 + (0.409 - 0.109i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.35 - 0.362i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 - 6.94T + 17T^{2} \)
19 \( 1 + (0.143 - 0.535i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 7.84iT - 23T^{2} \)
29 \( 1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.17 - 8.09i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.24 + 4.24i)T + 37iT^{2} \)
41 \( 1 + (-0.434 + 1.62i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.49 - 3.74i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.62 + 9.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.77 + 6.54i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.83 + 4.83i)T - 59iT^{2} \)
61 \( 1 + (-2.38 + 1.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.70 - 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.00 + 3.74i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (11.0 + 2.96i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.32 + 7.49i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.26 - 2.26i)T + 83iT^{2} \)
89 \( 1 + (-8.19 + 8.19i)T - 89iT^{2} \)
97 \( 1 + (-15.5 + 4.17i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.36648346435440446799639996998, −9.779257961386443231680130331878, −8.763590671820771914870434919470, −7.67105290742062207496685211061, −7.11180357967981309740861368223, −5.58970944157066647120318618327, −5.08546883372350978082137717969, −3.59837216828708798712286177804, −3.16093023559063937944225283475, −1.88005377085780152423043742668, 0.63281336510220199865086364382, 2.65512208269992215709428104241, 3.91636508491397667905006586375, 4.74264155984556377826003026185, 5.79713956357097166391077540452, 6.41777187267462444683420523624, 7.47324890889949423077415978085, 7.956371642874864386723624608986, 9.501137429665152466476073215983, 10.07074889336806679650448485261

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