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Properties

Label 2-819-91.24-c1-0-0
Degree $2$
Conductor $819$
Sign $-0.857 + 0.514i$
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  + (−2.39 + 0.640i)2-s + (3.57 − 2.06i)4-s + (1.06 + 0.286i)5-s + (0.327 + 2.62i)7-s + (−3.72 + 3.72i)8-s − 2.74·10-s + (−1.52 + 1.52i)11-s + (−3.47 − 0.970i)13-s + (−2.46 − 6.06i)14-s + (2.39 − 4.14i)16-s + (−1.59 − 2.76i)17-s + (−2.51 + 2.51i)19-s + (4.41 − 1.18i)20-s + (2.67 − 4.63i)22-s + (−0.620 − 0.358i)23-s + ⋯
L(s)  = 1  + (−1.69 + 0.453i)2-s + (1.78 − 1.03i)4-s + (0.478 + 0.128i)5-s + (0.123 + 0.992i)7-s + (−1.31 + 1.31i)8-s − 0.866·10-s + (−0.461 + 0.461i)11-s + (−0.963 − 0.269i)13-s + (−0.658 − 1.62i)14-s + (0.598 − 1.03i)16-s + (−0.387 − 0.670i)17-s + (−0.576 + 0.576i)19-s + (0.987 − 0.264i)20-s + (0.570 − 0.988i)22-s + (−0.129 − 0.0746i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.857 + 0.514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0275551 - 0.0994474i\)
\(L(\frac12)\) \(\approx\) \(0.0275551 - 0.0994474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.327 - 2.62i)T \)
13 \( 1 + (3.47 + 0.970i)T \)
good2 \( 1 + (2.39 - 0.640i)T + (1.73 - i)T^{2} \)
5 \( 1 + (-1.06 - 0.286i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.52 - 1.52i)T - 11iT^{2} \)
17 \( 1 + (1.59 + 2.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.51 - 2.51i)T - 19iT^{2} \)
23 \( 1 + (0.620 + 0.358i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.58 + 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.625 + 2.33i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.726 - 2.71i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.01 + 1.07i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.32 - 4.22i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.85 - 10.6i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.54 - 7.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.10 + 7.87i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (8.40 + 8.40i)T + 67iT^{2} \)
71 \( 1 + (-14.8 + 3.96i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.86 - 1.30i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.54 + 14.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.63 - 3.63i)T - 83iT^{2} \)
89 \( 1 + (-5.24 + 1.40i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.61 - 13.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.39110150581997319502242841805, −9.526356793987617288990700042034, −9.320519499934791232373539066815, −8.053873273803220147443511061636, −7.72943140652473677263680436820, −6.53865873921524919699587007868, −5.88527487009901315630524491779, −4.76245460537597799542730554794, −2.62676842237361448066839962094, −1.89565260564892161027683852918, 0.083229047096613557627619517790, 1.56163315431622877473869242194, 2.59283891191441742888041172597, 4.00646864062881807640932061939, 5.40460635302572982500166365965, 6.78202069052266216245186144811, 7.33030452055473000939638969843, 8.272107944195300046324660368854, 8.948534423874837515552973331720, 9.847093832340315495819179117313

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