↑

Properties

Label 2-3100-31.30-c2-0-64
Degree $2$
Conductor $3100$
Sign $0.755 + 0.654i$
Analytic cond. $84.4688$
Root an. cond. $9.19069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.25i·3-s − 6.21·7-s + 7.43·9-s − 14.0i·11-s + 19.3i·13-s − 28.3i·17-s + 12.6·19-s − 7.77i·21-s + 5.27i·23-s + 20.5i·27-s + 34.3i·29-s + (23.4 + 20.2i)31-s + 17.5·33-s − 34.6i·37-s − 24.1·39-s + ⋯
L(s)  = 1  + 0.417i·3-s − 0.888·7-s + 0.825·9-s − 1.27i·11-s + 1.48i·13-s − 1.66i·17-s + 0.665·19-s − 0.370i·21-s + 0.229i·23-s + 0.761i·27-s + 1.18i·29-s + (0.755 + 0.654i)31-s + 0.532·33-s − 0.935i·37-s − 0.619·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3100\)    =    \(2^{2} \cdot 5^{2} \cdot 31\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(84.4688\)
Root analytic conductor: \(9.19069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3100} (1301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3100,\ (\ :1),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.701945646\)
\(L(\frac12)\) \(\approx\) \(1.701945646\)
\(L(2)\) 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 \( 1 \)
31 \( 1 + (-23.4 - 20.2i)T \)
good3 \( 1 - 1.25iT - 9T^{2} \)
7 \( 1 + 6.21T + 49T^{2} \)
11 \( 1 + 14.0iT - 121T^{2} \)
13 \( 1 - 19.3iT - 169T^{2} \)
17 \( 1 + 28.3iT - 289T^{2} \)
19 \( 1 - 12.6T + 361T^{2} \)
23 \( 1 - 5.27iT - 529T^{2} \)
29 \( 1 - 34.3iT - 841T^{2} \)
37 \( 1 + 34.6iT - 1.36e3T^{2} \)
41 \( 1 + 59.5T + 1.68e3T^{2} \)
43 \( 1 - 1.25iT - 1.84e3T^{2} \)
47 \( 1 - 22.1T + 2.20e3T^{2} \)
53 \( 1 + 31.5iT - 2.80e3T^{2} \)
59 \( 1 + 41.9T + 3.48e3T^{2} \)
61 \( 1 + 46.8iT - 3.72e3T^{2} \)
67 \( 1 + 45.1T + 4.48e3T^{2} \)
71 \( 1 - 67.2T + 5.04e3T^{2} \)
73 \( 1 + 48.6iT - 5.32e3T^{2} \)
79 \( 1 + 151. iT - 6.24e3T^{2} \)
83 \( 1 + 87.7iT - 6.88e3T^{2} \)
89 \( 1 - 98.0iT - 7.92e3T^{2} \)
97 \( 1 + 100.T + 9.40e3T^{2} \)
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

−8.737611294957412684351029435631, −7.51045875035001146424735479066, −6.90031621671214962398257173648, −6.32026716509692718240751660849, −5.23142833704441453683803940092, −4.61506741369509424782618236844, −3.51840389244490458391918860771, −3.07819686087981391282624308591, −1.68000379848997505611717116638, −0.47126140340091443299099781471, 0.891076265340757477392414882284, 1.93143994174596908174003825514, 2.94387508280933525524838859313, 3.89457965842028280496487118782, 4.66436897902440226576328703833, 5.74203965099805458521420823973, 6.39833081410476904119450724577, 7.10761816174179562807951121144, 7.85476123231310186254271270123, 8.371556376532980632393594579513

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