Properties

Label 2-920-115.102-c1-0-17
Degree $2$
Conductor $920$
Sign $0.525 - 0.850i$
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  + (1.44 − 0.313i)3-s + (−0.625 + 2.14i)5-s + (2.34 − 1.27i)7-s + (−0.745 + 0.340i)9-s + (4.30 + 3.72i)11-s + (−2.02 + 3.71i)13-s + (−0.229 + 3.29i)15-s + (−2.53 − 1.89i)17-s + (−0.894 + 6.22i)19-s + (2.97 − 2.58i)21-s + (−1.71 − 4.47i)23-s + (−4.21 − 2.68i)25-s + (−4.51 + 3.37i)27-s + (2.66 − 0.383i)29-s + (5.88 − 3.78i)31-s + ⋯
L(s)  = 1  + (0.833 − 0.181i)3-s + (−0.279 + 0.960i)5-s + (0.885 − 0.483i)7-s + (−0.248 + 0.113i)9-s + (1.29 + 1.12i)11-s + (−0.562 + 1.03i)13-s + (−0.0591 + 0.850i)15-s + (−0.613 − 0.459i)17-s + (−0.205 + 1.42i)19-s + (0.650 − 0.563i)21-s + (−0.357 − 0.934i)23-s + (−0.843 − 0.537i)25-s + (−0.868 + 0.650i)27-s + (0.495 − 0.0711i)29-s + (1.05 − 0.679i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.525 - 0.850i$
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: \((2,\ 920,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80799 + 1.00796i\)
\(L(\frac12)\) \(\approx\) \(1.80799 + 1.00796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.625 - 2.14i)T \)
23 \( 1 + (1.71 + 4.47i)T \)
good3 \( 1 + (-1.44 + 0.313i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-2.34 + 1.27i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-4.30 - 3.72i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.02 - 3.71i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (2.53 + 1.89i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.894 - 6.22i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-2.66 + 0.383i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-5.88 + 3.78i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-7.38 - 2.75i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-2.17 + 4.75i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-0.0789 - 0.362i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-1.20 - 1.20i)T + 47iT^{2} \)
53 \( 1 + (-2.10 - 3.85i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (3.63 + 12.3i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-7.08 - 11.0i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (7.83 + 0.560i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-4.92 - 5.68i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.604 - 0.807i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-13.4 + 3.94i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.57 + 4.22i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (1.64 + 1.05i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (4.70 + 12.6i)T + (-73.3 + 63.5i)T^{2} \)
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

−10.08898390119698604394800435190, −9.420586509739012004793849964787, −8.394519260037376103819549677163, −7.70032334499870971077271970348, −6.97326793162708247714582293917, −6.20417938893876234831295260570, −4.47058376484558772644998590165, −4.05858865180410100880064355923, −2.59278868330643743502896702222, −1.79419859984344437637104611316, 0.941580001932799945534901163608, 2.45402403218806285309333320161, 3.56142065420764704987152040403, 4.55305206075056970890229591007, 5.45751021991608045576930653531, 6.43969508273424464064511828890, 7.88771600349218524719340485339, 8.353727742091459097240463358521, 8.994050970544421008926098643781, 9.531691555647967079870496660933

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