↑

Properties

Label 2-2548-91.88-c1-0-12
Degree $2$
Conductor $2548$
Sign $0.0882 - 0.996i$
Analytic cond. $20.3458$
Root an. cond. $4.51064$
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  + 0.901·3-s + (−1.46 + 0.844i)5-s − 2.18·9-s + 1.24i·11-s + (3.32 + 1.40i)13-s + (−1.31 + 0.760i)15-s + (1.80 + 3.12i)17-s − 7.44i·19-s + (4.18 − 7.25i)23-s + (−1.07 + 1.86i)25-s − 4.67·27-s + (3.09 + 5.35i)29-s + (1.13 + 0.655i)31-s + 1.11i·33-s + (8.90 + 5.14i)37-s + ⋯
L(s)  = 1  + 0.520·3-s + (−0.654 + 0.377i)5-s − 0.729·9-s + 0.373i·11-s + (0.921 + 0.389i)13-s + (−0.340 + 0.196i)15-s + (0.437 + 0.757i)17-s − 1.70i·19-s + (0.872 − 1.51i)23-s + (−0.214 + 0.372i)25-s − 0.899·27-s + (0.574 + 0.994i)29-s + (0.203 + 0.117i)31-s + 0.194i·33-s + (1.46 + 0.845i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0882 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0882 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2548\)    =    \(2^{2} \cdot 7^{2} \cdot 13\)
Sign: $0.0882 - 0.996i$
Analytic conductor: \(20.3458\)
Root analytic conductor: \(4.51064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2548} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2548,\ (\ :1/2),\ 0.0882 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.517652631\)
\(L(\frac12)\) \(\approx\) \(1.517652631\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 + (-3.32 - 1.40i)T \)
good3 \( 1 - 0.901T + 3T^{2} \)
5 \( 1 + (1.46 - 0.844i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 - 1.24iT - 11T^{2} \)
17 \( 1 + (-1.80 - 3.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + 7.44iT - 19T^{2} \)
23 \( 1 + (-4.18 + 7.25i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.09 - 5.35i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.13 - 0.655i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.90 - 5.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.40 + 0.812i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.78 - 8.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.3 - 5.97i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.95 - 8.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.11 - 2.37i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 5.42iT - 67T^{2} \)
71 \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.42 - 2.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.11 - 5.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.4iT - 83T^{2} \)
89 \( 1 + (-5.07 - 2.93i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.64 + 0.951i)T + (48.5 + 84.0i)T^{2} \)
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

−8.981158315617897972743367947729, −8.299695996547461934862333087189, −7.74310012214632597333207064360, −6.69327403599177315311277512227, −6.27039468897183818663361652199, −4.97146722952667479964321957357, −4.28030201706391351121815534542, −3.16406559988095320082308237669, −2.72818937293032993778919551405, −1.21267310887912172835476509290, 0.51491357818375203866685959292, 1.86319951507908110106897811400, 3.30482742730204041455474183558, 3.52131076619098897430076464264, 4.72518248671599701407547879885, 5.70132898645850933960019403197, 6.22845676982549574387484348311, 7.59929604297854121583726814672, 7.982619848136095400008271558942, 8.552979633980203168579277495570

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