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Properties

Label 2-819-91.34-c1-0-28
Degree $2$
Conductor $819$
Sign $0.389 - 0.920i$
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  + (1.45 + 1.45i)2-s + 2.21i·4-s + (2.01 − 2.01i)5-s + (−1.13 + 2.38i)7-s + (−0.311 + 0.311i)8-s + 5.84·10-s + (−0.451 + 0.451i)11-s + (3.40 − 1.19i)13-s + (−5.11 + 1.81i)14-s + 3.52·16-s + 4.32·17-s + (−3.40 + 3.40i)19-s + (4.46 + 4.46i)20-s − 1.31·22-s + 0.933i·23-s + ⋯
L(s)  = 1  + (1.02 + 1.02i)2-s + 1.10i·4-s + (0.900 − 0.900i)5-s + (−0.429 + 0.903i)7-s + (−0.109 + 0.109i)8-s + 1.84·10-s + (−0.136 + 0.136i)11-s + (0.943 − 0.330i)13-s + (−1.36 + 0.486i)14-s + 0.881·16-s + 1.04·17-s + (−0.780 + 0.780i)19-s + (0.997 + 0.997i)20-s − 0.279·22-s + 0.194i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.389 - 0.920i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.389 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59764 + 1.72093i\)
\(L(\frac12)\) \(\approx\) \(2.59764 + 1.72093i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 2.38i)T \)
13 \( 1 + (-3.40 + 1.19i)T \)
good2 \( 1 + (-1.45 - 1.45i)T + 2iT^{2} \)
5 \( 1 + (-2.01 + 2.01i)T - 5iT^{2} \)
11 \( 1 + (0.451 - 0.451i)T - 11iT^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + (3.40 - 3.40i)T - 19iT^{2} \)
23 \( 1 - 0.933iT - 23T^{2} \)
29 \( 1 + 6.33T + 29T^{2} \)
31 \( 1 + (-5.47 + 5.47i)T - 31iT^{2} \)
37 \( 1 + (-2.14 + 2.14i)T - 37iT^{2} \)
41 \( 1 + (1.81 - 1.81i)T - 41iT^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + (5.90 + 5.90i)T + 47iT^{2} \)
53 \( 1 + 3.36T + 53T^{2} \)
59 \( 1 + (-0.255 - 0.255i)T + 59iT^{2} \)
61 \( 1 + 7.78iT - 61T^{2} \)
67 \( 1 + (-7.28 - 7.28i)T + 67iT^{2} \)
71 \( 1 + (5.56 + 5.56i)T + 71iT^{2} \)
73 \( 1 + (8.86 + 8.86i)T + 73iT^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (4.30 - 4.30i)T - 83iT^{2} \)
89 \( 1 + (5.61 + 5.61i)T + 89iT^{2} \)
97 \( 1 + (0.236 - 0.236i)T - 97iT^{2} \)
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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

−10.10010648436024998308523227797, −9.528536233082389348937685634093, −8.464272829168614679725280803168, −7.82122148118195658285401801875, −6.45714158313176312458330858841, −5.82441394597496344625731456132, −5.43571192519228548608259278127, −4.34654451494194664985433277026, −3.20743431596971071316108314821, −1.59547213440903302772086212493, 1.43492746388035678306068519713, 2.68574889199557629581800307476, 3.46210904177348170202626013314, 4.37164961350038721074829213650, 5.56839385182950313993822362165, 6.37881752548711503325140889528, 7.21896313727740280547773827990, 8.482537567058087427605311675425, 9.760011750205749228919228916411, 10.36410918710567328961316720408

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