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Properties

Label 2-819-91.34-c1-0-6
Degree $2$
Conductor $819$
Sign $-0.883 + 0.467i$
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 + (−2.38 + 1.13i)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 + (−0.903 + 0.429i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.883 + 0.467i$
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: \((2,\ 819,\ (\ :1/2),\ -0.883 + 0.467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296413 - 1.19323i\)
\(L(\frac12)\) \(\approx\) \(0.296413 - 1.19323i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 - 1.13i)T \)
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

−10.91927064813008718279663428619, −9.782239785251444439877062260693, −8.957337172573460557289046937535, −7.51847032662866427139155337541, −7.22581362171204678111926191241, −6.46940224183846245525751426980, −5.50761387982464102128546403855, −4.51661881538502149258885018180, −3.57949489949324188420772296871, −2.66920047335868758302111588595, 0.41173112437466636605385971954, 2.15677309158733050738340063243, 3.43864856068506250095543820609, 4.05351159327926500917770749861, 4.94202246925828630468638065632, 5.82688721235211349945091530741, 7.24217477043472917709690373559, 7.963550257033950723083826721760, 9.125362611334592242647734507155, 9.951454384993887179585455122682

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