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Properties

Label 2-819-91.19-c1-0-3
Degree $2$
Conductor $819$
Sign $-0.783 - 0.621i$
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  + (−2.18 − 0.584i)2-s + (2.68 + 1.55i)4-s + (−2.27 + 0.609i)5-s + (−2.33 + 1.23i)7-s + (−1.76 − 1.76i)8-s + 5.31·10-s + (3.57 + 3.57i)11-s + (1.62 − 3.21i)13-s + (5.82 − 1.32i)14-s + (−0.288 − 0.499i)16-s + (2.64 − 4.57i)17-s + (2.98 + 2.98i)19-s + (−7.05 − 1.89i)20-s + (−5.70 − 9.87i)22-s + (−3.44 + 1.99i)23-s + ⋯
L(s)  = 1  + (−1.54 − 0.413i)2-s + (1.34 + 0.775i)4-s + (−1.01 + 0.272i)5-s + (−0.884 + 0.466i)7-s + (−0.623 − 0.623i)8-s + 1.68·10-s + (1.07 + 1.07i)11-s + (0.450 − 0.892i)13-s + (1.55 − 0.354i)14-s + (−0.0720 − 0.124i)16-s + (0.641 − 1.11i)17-s + (0.685 + 0.685i)19-s + (−1.57 − 0.422i)20-s + (−1.21 − 2.10i)22-s + (−0.719 + 0.415i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.783 - 0.621i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0591590 + 0.169911i\)
\(L(\frac12)\) \(\approx\) \(0.0591590 + 0.169911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 - 1.23i)T \)
13 \( 1 + (-1.62 + 3.21i)T \)
good2 \( 1 + (2.18 + 0.584i)T + (1.73 + i)T^{2} \)
5 \( 1 + (2.27 - 0.609i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.57 - 3.57i)T + 11iT^{2} \)
17 \( 1 + (-2.64 + 4.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.98 - 2.98i)T + 19iT^{2} \)
23 \( 1 + (3.44 - 1.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.565 - 0.978i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.40 - 5.23i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.37 + 5.13i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.61 - 1.50i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.65 - 5.57i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.95 + 7.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.538 - 0.931i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.18 - 8.16i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 3.15iT - 61T^{2} \)
67 \( 1 + (3.75 - 3.75i)T - 67iT^{2} \)
71 \( 1 + (12.9 + 3.47i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.56 - 0.418i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.31 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.21 - 7.21i)T + 83iT^{2} \)
89 \( 1 + (8.63 + 2.31i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.32 - 16.1i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.17780618065154896094017681541, −9.800215452368386510355432344286, −9.012798205945236817865358006427, −8.116217076840051977717867717929, −7.38206982995372518925645084779, −6.74834706719610010190261820247, −5.37309964797307595393392774471, −3.76903586514168655909770508845, −2.95903805415895469866990947152, −1.42286297276778716414771066410, 0.16922137019442910001816298959, 1.39875670428839711968798068989, 3.46704110100790281559161017074, 4.17969440747988526409788566271, 6.08553008982704955312777330862, 6.61319497581979963381544369273, 7.55516740465790240793800647626, 8.335316563112247300193735275387, 8.916788510969344996756449898616, 9.717165265803161109730146200217

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