↑

Properties

Label 2-819-91.81-c1-0-32
Degree $2$
Conductor $819$
Sign $0.521 + 0.853i$
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  + (0.532 − 0.922i)2-s + (0.432 + 0.748i)4-s + (−1.19 − 2.06i)5-s + (2.58 − 0.554i)7-s + 3.05·8-s − 2.53·10-s + 0.666·11-s + (2.19 + 2.85i)13-s + (0.866 − 2.68i)14-s + (0.761 − 1.31i)16-s + (0.707 + 1.22i)17-s − 3.56·19-s + (1.02 − 1.78i)20-s + (0.355 − 0.615i)22-s + (2.99 − 5.18i)23-s + ⋯
L(s)  = 1  + (0.376 − 0.652i)2-s + (0.216 + 0.374i)4-s + (−0.532 − 0.921i)5-s + (0.977 − 0.209i)7-s + 1.07·8-s − 0.802·10-s + 0.200·11-s + (0.609 + 0.792i)13-s + (0.231 − 0.716i)14-s + (0.190 − 0.329i)16-s + (0.171 + 0.297i)17-s − 0.817·19-s + (0.230 − 0.398i)20-s + (0.0757 − 0.131i)22-s + (0.623 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.521 + 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.521 + 0.853i$
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: \((2,\ 819,\ (\ :1/2),\ 0.521 + 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95695 - 1.09788i\)
\(L(\frac12)\) \(\approx\) \(1.95695 - 1.09788i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.58 + 0.554i)T \)
13 \( 1 + (-2.19 - 2.85i)T \)
good2 \( 1 + (-0.532 + 0.922i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1.19 + 2.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 0.666T + 11T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 3.56T + 19T^{2} \)
23 \( 1 + (-2.99 + 5.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.647 + 1.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.94 + 6.83i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.26 + 9.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.54 - 9.60i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.39 - 5.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.57 + 4.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 4.83T + 61T^{2} \)
67 \( 1 - 5.57T + 67T^{2} \)
71 \( 1 + (6.01 - 10.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.05 - 7.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.00 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.44T + 83T^{2} \)
89 \( 1 + (-0.910 + 1.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.88 - 13.6i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.60489207653417123467310628383, −8.953451390741396358839361569967, −8.541655321083111639574048123565, −7.63859713255855572851050944391, −6.76457797560235453658277655328, −5.34048205942185040839789370470, −4.26679096524346470213958865321, −3.99535819576429829390512536992, −2.35241562899336740396753506959, −1.22653396605298153167751040270, 1.47480998365317350508900396581, 2.94994559413422392535763999079, 4.17994495467554066508606451740, 5.19454463650319225169112120085, 6.01009976900591622040191678053, 6.91038285363472893915088296352, 7.67602931676890423076333545656, 8.309333368270705035275737307623, 9.608155431453984399806283305008, 10.60653812858453004591776599288

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