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Properties

Label 2-819-117.2-c1-0-29
Degree $2$
Conductor $819$
Sign $0.210 - 0.977i$
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  + (1.17 + 1.17i)2-s + (−1.63 − 0.572i)3-s + 0.774i·4-s + (−0.510 + 1.90i)5-s + (−1.25 − 2.59i)6-s + (−0.258 + 0.965i)7-s + (1.44 − 1.44i)8-s + (2.34 + 1.87i)9-s + (−2.84 + 1.64i)10-s + (3.71 − 3.71i)11-s + (0.443 − 1.26i)12-s + (0.444 − 3.57i)13-s + (−1.44 + 0.832i)14-s + (1.92 − 2.82i)15-s + 4.94·16-s + (−3.57 + 6.18i)17-s + ⋯
L(s)  = 1  + (0.832 + 0.832i)2-s + (−0.943 − 0.330i)3-s + 0.387i·4-s + (−0.228 + 0.851i)5-s + (−0.511 − 1.06i)6-s + (−0.0978 + 0.365i)7-s + (0.510 − 0.510i)8-s + (0.781 + 0.623i)9-s + (−0.899 + 0.519i)10-s + (1.12 − 1.12i)11-s + (0.127 − 0.365i)12-s + (0.123 − 0.992i)13-s + (−0.385 + 0.222i)14-s + (0.496 − 0.728i)15-s + 1.23·16-s + (−0.866 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.210 - 0.977i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43644 + 1.16053i\)
\(L(\frac12)\) \(\approx\) \(1.43644 + 1.16053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 + 0.572i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.444 + 3.57i)T \)
good2 \( 1 + (-1.17 - 1.17i)T + 2iT^{2} \)
5 \( 1 + (0.510 - 1.90i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.71 + 3.71i)T - 11iT^{2} \)
17 \( 1 + (3.57 - 6.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.43 - 5.35i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.59 - 4.49i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.68iT - 29T^{2} \)
31 \( 1 + (-9.58 - 2.56i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.30 + 4.88i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.01 + 1.07i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.65 - 2.68i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.08 - 7.78i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 8.77iT - 53T^{2} \)
59 \( 1 + (-5.41 + 5.41i)T - 59iT^{2} \)
61 \( 1 + (2.48 + 4.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.02 + 11.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.5 + 2.81i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (1.53 + 1.53i)T + 73iT^{2} \)
79 \( 1 + (4.30 - 7.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.32 + 1.69i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (11.6 + 3.13i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.98 + 1.60i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.69845839692615360647988354269, −9.790998058914605417895438588654, −8.322483557278990534241402420388, −7.57187630633089209454597446635, −6.42475870181881983613381346827, −6.19795688919698341080215737704, −5.49301957940558882645138984126, −4.20028414458431337612428705559, −3.35701670986286050263047004090, −1.34657874434673693882591503508, 0.950943512894612541941334967874, 2.41623358342841532308417676384, 4.09917578970434803282539203636, 4.49845204813721041566654915072, 5.01402909980175510679551008010, 6.57822817605246925209545710204, 7.09286065387809511085977780523, 8.592319624706875090670194597662, 9.484322126602045499293726260654, 10.17932720169616475149701019461

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