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Properties

Label 2-819-91.5-c1-0-32
Degree $2$
Conductor $819$
Sign $0.903 - 0.429i$
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  + (0.411 + 1.53i)2-s + (−0.454 + 0.262i)4-s + (0.0769 − 0.0206i)5-s + (1.52 − 2.16i)7-s + (1.65 + 1.65i)8-s + (0.0632 + 0.109i)10-s + (1.09 − 4.08i)11-s + (0.565 − 3.56i)13-s + (3.94 + 1.44i)14-s + (−2.38 + 4.13i)16-s + (−2.90 − 5.02i)17-s + (5.11 − 1.36i)19-s + (−0.0295 + 0.0295i)20-s + 6.71·22-s + (−0.755 − 0.436i)23-s + ⋯
L(s)  = 1  + (0.290 + 1.08i)2-s + (−0.227 + 0.131i)4-s + (0.0343 − 0.00921i)5-s + (0.575 − 0.818i)7-s + (0.585 + 0.585i)8-s + (0.0200 + 0.0346i)10-s + (0.329 − 1.23i)11-s + (0.156 − 0.987i)13-s + (1.05 + 0.386i)14-s + (−0.596 + 1.03i)16-s + (−0.704 − 1.21i)17-s + (1.17 − 0.314i)19-s + (−0.00661 + 0.00661i)20-s + 1.43·22-s + (−0.157 − 0.0909i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09273 + 0.472441i\)
\(L(\frac12)\) \(\approx\) \(2.09273 + 0.472441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 2.16i)T \)
13 \( 1 + (-0.565 + 3.56i)T \)
good2 \( 1 + (-0.411 - 1.53i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-0.0769 + 0.0206i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.09 + 4.08i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.90 + 5.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.11 + 1.36i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.755 + 0.436i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.362T + 29T^{2} \)
31 \( 1 + (0.361 - 1.34i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.76 - 1.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.70 - 7.70i)T + 41iT^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 + (-0.748 - 2.79i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.26 - 9.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.14 - 0.573i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.63 - 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.76 - 2.61i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.65 - 3.65i)T - 71iT^{2} \)
73 \( 1 + (11.4 + 3.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.27 - 7.40i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.91 + 4.91i)T + 83iT^{2} \)
89 \( 1 + (-2.08 - 7.78i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-6.04 - 6.04i)T + 97iT^{2} \)
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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

−10.37330005275736553305912144592, −9.259480164866022199538657436409, −8.264454942378279166717094715641, −7.59139071908922755621592686326, −6.91998300744431535274001364278, −5.85741272790435542618989068015, −5.22327503842637231099751743127, −4.18998739717300113268561560486, −2.90158160777941425153289665557, −1.07107511928452544266027754322, 1.74508978569285280431455212444, 2.21306854894448587520271443696, 3.76640485857880027331620245373, 4.42333665706808388839507812761, 5.58144452076829273239499621441, 6.74649226415913461574519337070, 7.59545806351652673188018095258, 8.705778338248643689485339026553, 9.554499331894167777985885834861, 10.26597745887621159059446326105

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