↑

Properties

Label 2-920-115.102-c1-0-13
Degree $2$
Conductor $920$
Sign $0.767 - 0.641i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (−3.05 + 0.664i)3-s + (2.09 − 0.785i)5-s + (−1.51 + 0.825i)7-s + (6.17 − 2.81i)9-s + (1.45 + 1.25i)11-s + (−1.50 + 2.76i)13-s + (−5.87 + 3.79i)15-s + (−5.05 − 3.78i)17-s + (0.770 − 5.36i)19-s + (4.07 − 3.52i)21-s + (3.96 + 2.69i)23-s + (3.76 − 3.28i)25-s + (−9.48 + 7.09i)27-s + (7.28 − 1.04i)29-s + (−4.63 + 2.98i)31-s + ⋯
L(s)  = 1  + (−1.76 + 0.383i)3-s + (0.936 − 0.351i)5-s + (−0.571 + 0.312i)7-s + (2.05 − 0.939i)9-s + (0.437 + 0.379i)11-s + (−0.418 + 0.765i)13-s + (−1.51 + 0.979i)15-s + (−1.22 − 0.917i)17-s + (0.176 − 1.23i)19-s + (0.888 − 0.770i)21-s + (0.826 + 0.562i)23-s + (0.753 − 0.657i)25-s + (−1.82 + 1.36i)27-s + (1.35 − 0.194i)29-s + (−0.833 + 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.767 - 0.641i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.767 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.844738 + 0.306681i\)
\(L(\frac12)\) \(\approx\) \(0.844738 + 0.306681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-2.09 + 0.785i)T \)
23 \( 1 + (-3.96 - 2.69i)T \)
good3 \( 1 + (3.05 - 0.664i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (1.51 - 0.825i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-1.45 - 1.25i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.50 - 2.76i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (5.05 + 3.78i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.770 + 5.36i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-7.28 + 1.04i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (4.63 - 2.98i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-6.47 - 2.41i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (2.27 - 4.98i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.26 + 5.80i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-9.54 - 9.54i)T + 47iT^{2} \)
53 \( 1 + (-2.20 - 4.03i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-1.46 - 4.97i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-6.07 - 9.45i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-10.4 - 0.747i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-0.656 - 0.757i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.00 - 2.68i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-6.98 + 2.05i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-5.02 + 13.4i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-2.34 - 1.50i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-0.180 - 0.484i)T + (-73.3 + 63.5i)T^{2} \)
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

−10.20560289309898601115408818715, −9.305526513174894508965938949798, −9.133697794975488337122641453056, −7.00839348066059453841921401952, −6.69778134492477906407004020019, −5.81826911330903159194337358789, −4.89337871819457671831528726144, −4.45848137249266843082727958525, −2.56113375541729132217755435945, −0.963268188400159443287886563864, 0.71905886489156025718362944929, 2.11702690453007158539224358387, 3.77640514683881412413022421864, 5.05174570511667880603479210037, 5.77627921160404505138816686697, 6.53367608287189426880867609633, 6.89268897871005648542459790247, 8.199343652788720804403662045379, 9.487747209206568243656090568295, 10.31491477580145113865175119162

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