↑

Properties

Label 2-819-117.103-c1-0-25
Degree $2$
Conductor $819$
Sign $0.206 - 0.978i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.56 − 0.901i)2-s + (−1.24 + 1.20i)3-s + (0.624 − 1.08i)4-s + (2.25 + 1.30i)5-s + (−0.850 + 3.00i)6-s + (−0.866 + 0.5i)7-s + 1.35i·8-s + (0.0843 − 2.99i)9-s + 4.69·10-s + (−4.12 + 2.38i)11-s + (0.530 + 2.09i)12-s + (0.606 + 3.55i)13-s + (−0.901 + 1.56i)14-s + (−4.37 + 1.10i)15-s + (2.46 + 4.27i)16-s − 0.0663·17-s + ⋯
L(s)  = 1  + (1.10 − 0.637i)2-s + (−0.716 + 0.697i)3-s + (0.312 − 0.541i)4-s + (1.00 + 0.582i)5-s + (−0.347 + 1.22i)6-s + (−0.327 + 0.188i)7-s + 0.478i·8-s + (0.0281 − 0.999i)9-s + 1.48·10-s + (−1.24 + 0.718i)11-s + (0.153 + 0.605i)12-s + (0.168 + 0.985i)13-s + (−0.240 + 0.417i)14-s + (−1.12 + 0.285i)15-s + (0.617 + 1.06i)16-s − 0.0160·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59684 + 1.29483i\)
\(L(\frac12)\) \(\approx\) \(1.59684 + 1.29483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.24 - 1.20i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.606 - 3.55i)T \)
good2 \( 1 + (-1.56 + 0.901i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.25 - 1.30i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.12 - 2.38i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.0663T + 17T^{2} \)
19 \( 1 + 4.42iT - 19T^{2} \)
23 \( 1 + (2.02 - 3.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.47 + 4.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.99 + 1.15i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.25iT - 37T^{2} \)
41 \( 1 + (-10.3 - 5.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.32 - 10.9i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.37 + 0.793i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.13T + 53T^{2} \)
59 \( 1 + (8.75 + 5.05i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.00 + 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-9.10 - 5.25i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.35iT - 71T^{2} \)
73 \( 1 + 11.0iT - 73T^{2} \)
79 \( 1 + (-0.0594 - 0.103i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-15.1 + 8.75i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.30iT - 89T^{2} \)
97 \( 1 + (-11.7 + 6.76i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.66220072269448741909283727501, −9.758007596038144843638565596457, −9.254498572733412894200461528556, −7.69502272734773381160599402998, −6.41804451284649265687238252494, −5.83440684870801422056094870011, −4.93305436396932462440084072608, −4.21147246259161678710593118045, −2.96117065412847347809032626508, −2.12030477972798979632063714967, 0.75719116728144128277936688250, 2.42422209771394988747727350525, 3.86200216732175203956670345669, 5.28380306122280431500208339101, 5.56943526500747244544044254605, 6.11070761871925548697741726981, 7.24208624573231824929674117710, 7.982483632301324363265215103987, 9.161691702135886185271751158486, 10.47832515421274515976246392232

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