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Properties

Label 2-819-117.2-c1-0-55
Degree $2$
Conductor $819$
Sign $0.392 + 0.919i$
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  + (0.861 + 0.861i)2-s + (−1.45 − 0.932i)3-s − 0.514i·4-s + (−0.395 + 1.47i)5-s + (−0.454 − 2.06i)6-s + (0.258 − 0.965i)7-s + (2.16 − 2.16i)8-s + (1.26 + 2.72i)9-s + (−1.61 + 0.931i)10-s + (−0.280 + 0.280i)11-s + (−0.479 + 0.750i)12-s + (−2.94 − 2.08i)13-s + (1.05 − 0.609i)14-s + (1.95 − 1.78i)15-s + 2.70·16-s + (1.67 − 2.90i)17-s + ⋯
L(s)  = 1  + (0.609 + 0.609i)2-s + (−0.842 − 0.538i)3-s − 0.257i·4-s + (−0.176 + 0.659i)5-s + (−0.185 − 0.841i)6-s + (0.0978 − 0.365i)7-s + (0.766 − 0.766i)8-s + (0.420 + 0.907i)9-s + (−0.510 + 0.294i)10-s + (−0.0845 + 0.0845i)11-s + (−0.138 + 0.216i)12-s + (−0.816 − 0.577i)13-s + (0.282 − 0.162i)14-s + (0.504 − 0.460i)15-s + 0.676·16-s + (0.406 − 0.703i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.392 + 0.919i$
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: \((2,\ 819,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12903 - 0.745646i\)
\(L(\frac12)\) \(\approx\) \(1.12903 - 0.745646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.45 + 0.932i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
13 \( 1 + (2.94 + 2.08i)T \)
good2 \( 1 + (-0.861 - 0.861i)T + 2iT^{2} \)
5 \( 1 + (0.395 - 1.47i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.280 - 0.280i)T - 11iT^{2} \)
17 \( 1 + (-1.67 + 2.90i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.227 + 0.847i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.44 + 2.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.38iT - 29T^{2} \)
31 \( 1 + (10.0 + 2.69i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.16 + 4.33i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.76 + 1.54i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.45 + 0.837i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.35 + 8.80i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 1.65iT - 53T^{2} \)
59 \( 1 + (-1.50 + 1.50i)T - 59iT^{2} \)
61 \( 1 + (-4.12 - 7.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.199 + 0.743i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-4.37 + 1.17i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.05 - 3.05i)T + 73iT^{2} \)
79 \( 1 + (1.00 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.30 + 0.617i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-11.1 - 2.97i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.98 + 0.531i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.34599584224390020958871368390, −9.427839904052169844576557254172, −7.74998252956493500922440100215, −7.28692049971266690182650272910, −6.62908495994846021506474196477, −5.61645732028276150155690557457, −5.04323573669271517808784663426, −3.95008130892871426475745877688, −2.34852196977209824779198910416, −0.63232681336798341028447303237, 1.58314856555913055884155542421, 3.14634241027406143966106753232, 4.13762883283795435022617817691, 4.92837841484503315963897025959, 5.54782435328666020913238237887, 6.83495068857777031703201183837, 7.86493745175666023314844777588, 8.877008015813843787039472820784, 9.634449776672532887247063168869, 10.77646402798315648768646667111

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