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Properties

Label 2-819-117.2-c1-0-25
Degree $2$
Conductor $819$
Sign $-0.968 - 0.248i$
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  + (1.04 + 1.04i)2-s + (0.533 + 1.64i)3-s + 0.171i·4-s + (−0.558 + 2.08i)5-s + (−1.16 + 2.27i)6-s + (−0.258 + 0.965i)7-s + (1.90 − 1.90i)8-s + (−2.42 + 1.75i)9-s + (−2.75 + 1.58i)10-s + (−2.65 + 2.65i)11-s + (−0.282 + 0.0914i)12-s + (0.388 + 3.58i)13-s + (−1.27 + 0.736i)14-s + (−3.73 + 0.192i)15-s + 4.31·16-s + (2.62 − 4.55i)17-s + ⋯
L(s)  = 1  + (0.736 + 0.736i)2-s + (0.308 + 0.951i)3-s + 0.0855i·4-s + (−0.249 + 0.932i)5-s + (−0.473 + 0.927i)6-s + (−0.0978 + 0.365i)7-s + (0.673 − 0.673i)8-s + (−0.809 + 0.586i)9-s + (−0.870 + 0.502i)10-s + (−0.801 + 0.801i)11-s + (−0.0814 + 0.0263i)12-s + (0.107 + 0.994i)13-s + (−0.341 + 0.196i)14-s + (−0.963 + 0.0497i)15-s + 1.07·16-s + (0.637 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.968 - 0.248i$
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: \((2,\ 819,\ (\ :1/2),\ -0.968 - 0.248i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.268140 + 2.12500i\)
\(L(\frac12)\) \(\approx\) \(0.268140 + 2.12500i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.533 - 1.64i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (-0.388 - 3.58i)T \)
good2 \( 1 + (-1.04 - 1.04i)T + 2iT^{2} \)
5 \( 1 + (0.558 - 2.08i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.65 - 2.65i)T - 11iT^{2} \)
17 \( 1 + (-2.62 + 4.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.17 + 4.37i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.78 - 4.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.85iT - 29T^{2} \)
31 \( 1 + (4.07 + 1.09i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.408 + 1.52i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-11.7 + 3.14i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.30 - 0.751i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.54 - 9.48i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + (-6.93 + 6.93i)T - 59iT^{2} \)
61 \( 1 + (4.61 + 7.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.13 + 4.21i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.81 + 1.02i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.46 - 2.46i)T + 73iT^{2} \)
79 \( 1 + (2.63 - 4.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-13.5 + 3.63i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.206 + 0.0554i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-14.4 - 3.87i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.72328278140988197335098703290, −9.607431349597873442968896344061, −9.244112093811654249262504748185, −7.65224132923912412231009610619, −7.22970986690719124668394552889, −6.12953738686784812072124484591, −5.19853048883636852430005642032, −4.51001131273027041695915948140, −3.46631442730855903339345544997, −2.39084977024573288410669650596, 0.808123455308238691770864706335, 2.16716127118544464271832969046, 3.29672600331664235591173260309, 4.08935277230652789410770584576, 5.38641655255030159630847722182, 6.06176203573563806365750697171, 7.58345124650799038815823050371, 8.215505881808622179972453414883, 8.541361772912770891226018816129, 10.18039258581629174338756094378

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