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Properties

Label 2-819-91.81-c1-0-19
Degree $2$
Conductor $819$
Sign $0.991 + 0.129i$
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  + (−1.21 + 2.10i)2-s + (−1.96 − 3.39i)4-s + (0.613 + 1.06i)5-s + (−2.37 − 1.17i)7-s + 4.68·8-s − 2.98·10-s − 3.49·11-s + (−2.87 + 2.17i)13-s + (5.35 − 3.57i)14-s + (−1.77 + 3.07i)16-s + (2.26 + 3.92i)17-s + 1.35·19-s + (2.40 − 4.17i)20-s + (4.25 − 7.36i)22-s + (−0.336 + 0.583i)23-s + ⋯
L(s)  = 1  + (−0.860 + 1.49i)2-s + (−0.981 − 1.69i)4-s + (0.274 + 0.475i)5-s + (−0.896 − 0.443i)7-s + 1.65·8-s − 0.945·10-s − 1.05·11-s + (−0.797 + 0.603i)13-s + (1.43 − 0.954i)14-s + (−0.444 + 0.769i)16-s + (0.548 + 0.950i)17-s + 0.310·19-s + (0.538 − 0.933i)20-s + (0.906 − 1.56i)22-s + (−0.0701 + 0.121i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397361 - 0.0259338i\)
\(L(\frac12)\) \(\approx\) \(0.397361 - 0.0259338i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.37 + 1.17i)T \)
13 \( 1 + (2.87 - 2.17i)T \)
good2 \( 1 + (1.21 - 2.10i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-0.613 - 1.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 3.49T + 11T^{2} \)
17 \( 1 + (-2.26 - 3.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + (0.336 - 0.583i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.64 + 4.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.99 + 8.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.54 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.61 + 6.25i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.48 + 7.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.58 + 4.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.95 + 8.57i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.401 - 0.695i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.64T + 61T^{2} \)
67 \( 1 + 2.12T + 67T^{2} \)
71 \( 1 + (-2.52 + 4.37i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.04 - 10.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.90 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + (1.55 - 2.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.59 - 6.22i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.13568935115295754333126751234, −9.335063424888652843123500487159, −8.310346171992029541455719372630, −7.54158587208016676194642800259, −6.91368314547817004256564397565, −6.08207167998767954222374692858, −5.37858096625700921117190964173, −4.01759371828007333281900763587, −2.45176276020761611991307544419, −0.28988813346988428727932095573, 1.18727514358592665360195438869, 2.82133644036110197068371648257, 3.02996196399112021815942711699, 4.75178002040714633188717433101, 5.61973250323274544613644785075, 7.11932525974850867191092317132, 8.069378161577283548842801538590, 8.900131465283721599947366379599, 9.690766039799210910854115197117, 10.05787247542563981003335818521

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