↑

Properties

Label 2-2217-2217.689-c0-0-0
Degree $2$
Conductor $2217$
Sign $0.975 - 0.219i$
Analytic cond. $1.10642$
Root an. cond. $1.05186$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.988 − 0.152i)3-s + (−0.543 − 0.839i)4-s + (−0.821 + 1.51i)7-s + (0.953 − 0.301i)9-s + (−0.665 − 0.746i)12-s + (1.72 + 0.693i)13-s + (−0.409 + 0.912i)16-s + (−0.321 + 1.16i)19-s + (−0.581 + 1.61i)21-s + (0.953 + 0.301i)25-s + (0.896 − 0.443i)27-s + (1.71 − 0.131i)28-s + (−1.76 − 0.712i)31-s + (−0.771 − 0.636i)36-s + (0.796 − 1.77i)37-s + ⋯
L(s)  = 1  + (0.988 − 0.152i)3-s + (−0.543 − 0.839i)4-s + (−0.821 + 1.51i)7-s + (0.953 − 0.301i)9-s + (−0.665 − 0.746i)12-s + (1.72 + 0.693i)13-s + (−0.409 + 0.912i)16-s + (−0.321 + 1.16i)19-s + (−0.581 + 1.61i)21-s + (0.953 + 0.301i)25-s + (0.896 − 0.443i)27-s + (1.71 − 0.131i)28-s + (−1.76 − 0.712i)31-s + (−0.771 − 0.636i)36-s + (0.796 − 1.77i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2217 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2217\)    =    \(3 \cdot 739\)
Sign: $0.975 - 0.219i$
Analytic conductor: \(1.10642\)
Root analytic conductor: \(1.05186\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2217} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2217,\ (\ :0),\ 0.975 - 0.219i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.443017749\)
\(L(\frac12)\) \(\approx\) \(1.443017749\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.988 + 0.152i)T \)
739 \( 1 + (0.665 + 0.746i)T \)
good2 \( 1 + (0.543 + 0.839i)T^{2} \)
5 \( 1 + (-0.953 - 0.301i)T^{2} \)
7 \( 1 + (0.821 - 1.51i)T + (-0.543 - 0.839i)T^{2} \)
11 \( 1 + (-0.0383 + 0.999i)T^{2} \)
13 \( 1 + (-1.72 - 0.693i)T + (0.720 + 0.693i)T^{2} \)
17 \( 1 + (-0.477 + 0.878i)T^{2} \)
19 \( 1 + (0.321 - 1.16i)T + (-0.859 - 0.511i)T^{2} \)
23 \( 1 + (0.543 + 0.839i)T^{2} \)
29 \( 1 + (-0.190 - 0.981i)T^{2} \)
31 \( 1 + (1.76 + 0.712i)T + (0.720 + 0.693i)T^{2} \)
37 \( 1 + (-0.796 + 1.77i)T + (-0.665 - 0.746i)T^{2} \)
41 \( 1 + (0.927 - 0.373i)T^{2} \)
43 \( 1 + (-1.08 - 0.537i)T + (0.606 + 0.795i)T^{2} \)
47 \( 1 + (0.997 - 0.0765i)T^{2} \)
53 \( 1 + (0.665 + 0.746i)T^{2} \)
59 \( 1 + (-0.606 + 0.795i)T^{2} \)
61 \( 1 + (0.276 + 0.769i)T + (-0.771 + 0.636i)T^{2} \)
67 \( 1 + (1.03 - 0.328i)T + (0.817 - 0.575i)T^{2} \)
71 \( 1 + (0.997 + 0.0765i)T^{2} \)
73 \( 1 + (-1.55 - 1.09i)T + (0.338 + 0.941i)T^{2} \)
79 \( 1 + (0.930 + 0.217i)T + (0.896 + 0.443i)T^{2} \)
83 \( 1 + (-0.896 - 0.443i)T^{2} \)
89 \( 1 + (-0.720 + 0.693i)T^{2} \)
97 \( 1 + (0.635 - 1.16i)T + (-0.543 - 0.839i)T^{2} \)
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

−9.179026828233050850951072243376, −8.788959508559299929211301606617, −7.999295878228419088943255671713, −6.79156143329570394096278052855, −5.99881122729415448979380332957, −5.59160568491961489360472494903, −4.17725571491003869308100444411, −3.55983546879645450451772381687, −2.37137205069054878767244008029, −1.48882136801613161948651886485, 1.05077917501829685058071191161, 2.86409747802897390980146836679, 3.46839937879753887067563851000, 4.05902605005625265949446644850, 4.85912692118631844431905717817, 6.35549262466929298350002360418, 7.12592379680614004629260400174, 7.68584524032260685775325824919, 8.586293956313809045253743887498, 8.975991320509914044848560155941

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