Properties

Label 2-819-91.59-c1-0-39
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: $\chi_{819} (514, \cdot )$
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

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

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