↑

Properties

Label 2-440-11.3-c1-0-4
Degree $2$
Conductor $440$
Sign $0.996 - 0.0791i$
Analytic cond. $3.51341$
Root an. cond. $1.87441$
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  + (−0.320 − 0.987i)3-s + (0.809 + 0.587i)5-s + (−0.804 + 2.47i)7-s + (1.55 − 1.13i)9-s + (1.34 + 3.03i)11-s + (2.32 − 1.69i)13-s + (0.320 − 0.987i)15-s + (−0.655 − 0.476i)17-s + (1.59 + 4.89i)19-s + 2.70·21-s + 4.98·23-s + (0.309 + 0.951i)25-s + (−4.13 − 3.00i)27-s + (2.07 − 6.39i)29-s + (3.35 − 2.43i)31-s + ⋯
L(s)  = 1  + (−0.185 − 0.569i)3-s + (0.361 + 0.262i)5-s + (−0.303 + 0.935i)7-s + (0.518 − 0.376i)9-s + (0.405 + 0.914i)11-s + (0.645 − 0.469i)13-s + (0.0828 − 0.254i)15-s + (−0.158 − 0.115i)17-s + (0.365 + 1.12i)19-s + 0.589·21-s + 1.03·23-s + (0.0618 + 0.190i)25-s + (−0.795 − 0.577i)27-s + (0.386 − 1.18i)29-s + (0.601 − 0.437i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(440\)    =    \(2^{3} \cdot 5 \cdot 11\)
Sign: $0.996 - 0.0791i$
Analytic conductor: \(3.51341\)
Root analytic conductor: \(1.87441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{440} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 440,\ (\ :1/2),\ 0.996 - 0.0791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47306 + 0.0583728i\)
\(L(\frac12)\) \(\approx\) \(1.47306 + 0.0583728i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.34 - 3.03i)T \)
good3 \( 1 + (0.320 + 0.987i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (0.804 - 2.47i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.32 + 1.69i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.655 + 0.476i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.59 - 4.89i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 + (-2.07 + 6.39i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.35 + 2.43i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.25 + 3.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.01 - 9.28i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.92T + 43T^{2} \)
47 \( 1 + (0.299 + 0.920i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.61 - 2.63i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.10 + 6.47i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.11 - 1.53i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 + (-1.14 - 0.830i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.06 - 12.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.60 - 3.34i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (6.44 + 4.68i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 + (-2.42 + 1.75i)T + (29.9 - 92.2i)T^{2} \)
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

−11.33003325832233866756983873114, −10.00436385600161714936294983691, −9.530431371289354655895117763155, −8.373267522889257513674793719719, −7.32315743650999671862697900307, −6.38428728022148683592252574596, −5.72947300806673650158330341365, −4.27994993262678306268049142793, −2.84150416631865879019943857595, −1.47742410169180444556460482802, 1.20207517374725923348083962320, 3.21843196969202612812043662965, 4.31182423945343841111601969022, 5.21011812329894443270739739007, 6.50151777099592812187492996946, 7.25087612132914482128927006721, 8.661009795778583358664776464893, 9.291697995596693535245654741521, 10.41656800250899068556788620395, 10.84949737089870565443656371199

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