↑

Properties

Label 2-920-5.4-c1-0-18
Degree $2$
Conductor $920$
Sign $0.894 - 0.447i$
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  + 2i·3-s + (2 − i)5-s − 3i·7-s − 9-s + 6i·13-s + (2 + 4i)15-s − 7i·17-s + 4·19-s + 6·21-s + i·23-s + (3 − 4i)25-s + 4i·27-s + 9·29-s − 3·31-s + (−3 − 6i)35-s + ⋯
L(s)  = 1  + 1.15i·3-s + (0.894 − 0.447i)5-s − 1.13i·7-s − 0.333·9-s + 1.66i·13-s + (0.516 + 1.03i)15-s − 1.69i·17-s + 0.917·19-s + 1.30·21-s + 0.208i·23-s + (0.600 − 0.800i)25-s + 0.769i·27-s + 1.67·29-s − 0.538·31-s + (−0.507 − 1.01i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90184 + 0.448965i\)
\(L(\frac12)\) \(\approx\) \(1.90184 + 0.448965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + i)T \)
23 \( 1 - iT \)
good3 \( 1 - 2iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 7iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 7iT - 53T^{2} \)
59 \( 1 + 9T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 13iT - 67T^{2} \)
71 \( 1 + 13T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 - 11iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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

−9.970993032740773998571020453231, −9.400650950863595796719640671996, −8.976314130652918751785312420792, −7.45642627460584773199300115106, −6.80303444331832880486229207689, −5.56877495154739735307388811593, −4.61023167475323989110401084390, −4.18706038023156161642051343076, −2.79929210880775103041963617438, −1.19881395260270307002118936162, 1.27796063889217452104315042610, 2.36153445555691264568359490626, 3.21039594208399749656021357050, 5.06330987539490000046857100460, 6.03808524500804080236427592142, 6.30656364495163437392375721977, 7.56119391130793280560037088721, 8.196658317727756488962915606710, 9.078263576763576428455897395614, 10.16740567927594096190764852702

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