↑

Properties

Label 2-5292-63.41-c1-0-21
Degree $2$
Conductor $5292$
Sign $0.762 - 0.647i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
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  + (−0.0382 + 0.0661i)5-s + (4.66 − 2.69i)11-s + (4.60 + 2.65i)13-s + 3.78·17-s + 5.01i·19-s + (−2.02 − 1.16i)23-s + (2.49 + 4.32i)25-s + (−8.84 + 5.10i)29-s + (4.97 + 2.87i)31-s − 0.708·37-s + (3.29 − 5.71i)41-s + (0.716 + 1.24i)43-s + (−1.46 − 2.53i)47-s + 12.1i·53-s + 0.411i·55-s + ⋯
L(s)  = 1  + (−0.0170 + 0.0295i)5-s + (1.40 − 0.811i)11-s + (1.27 + 0.737i)13-s + 0.917·17-s + 1.14i·19-s + (−0.422 − 0.243i)23-s + (0.499 + 0.865i)25-s + (−1.64 + 0.948i)29-s + (0.893 + 0.516i)31-s − 0.116·37-s + (0.515 − 0.892i)41-s + (0.109 + 0.189i)43-s + (−0.213 − 0.369i)47-s + 1.66i·53-s + 0.0554i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.762 - 0.647i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.762 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.351495873\)
\(L(\frac12)\) \(\approx\) \(2.351495873\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.0382 - 0.0661i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.66 + 2.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.60 - 2.65i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.78T + 17T^{2} \)
19 \( 1 - 5.01iT - 19T^{2} \)
23 \( 1 + (2.02 + 1.16i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.84 - 5.10i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.97 - 2.87i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.708T + 37T^{2} \)
41 \( 1 + (-3.29 + 5.71i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.716 - 1.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.46 + 2.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.1iT - 53T^{2} \)
59 \( 1 + (0.289 - 0.502i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.40 - 1.38i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.63 - 4.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.32iT - 71T^{2} \)
73 \( 1 + 7.12iT - 73T^{2} \)
79 \( 1 + (0.469 + 0.812i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.49 + 11.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.03T + 89T^{2} \)
97 \( 1 + (6.18 - 3.56i)T + (48.5 - 84.0i)T^{2} \)
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

−8.398665839867747260710922149970, −7.52793132673350320327283777053, −6.80905305246126360290545794035, −5.96958453118929446888704097836, −5.70343672370490021923972216160, −4.42501145325039294388601460184, −3.66089195456943223307269653438, −3.27474932833054775218350834614, −1.68784478326644123338110931358, −1.14545396999576397336107769666, 0.73084941229013101757483295929, 1.66098003777611759730792610932, 2.77119498262484040816888055033, 3.74753948604036406162549896238, 4.26663994482285856286612857631, 5.23419248057651134472347178144, 6.09163169230837991289732316894, 6.56788407030990709862597287592, 7.43196359253513746055552744287, 8.127737205495906890001278032099

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