↑

Properties

Label 2-5292-63.38-c1-0-38
Degree $2$
Conductor $5292$
Sign $-0.999 + 0.00584i$
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  + (1.37 − 2.37i)5-s + (0.362 − 0.209i)11-s + (−1.32 + 0.765i)13-s + (−1.95 + 3.38i)17-s + (5.11 − 2.95i)19-s + (−7.72 − 4.46i)23-s + (−1.26 − 2.18i)25-s + (−6.00 − 3.46i)29-s + 3.52i·31-s + (−4.54 − 7.87i)37-s + (1.06 + 1.84i)41-s + (−5.77 + 10.0i)43-s − 1.77·47-s + (3.39 + 1.96i)53-s − 1.14i·55-s + ⋯
L(s)  = 1  + (0.613 − 1.06i)5-s + (0.109 − 0.0630i)11-s + (−0.367 + 0.212i)13-s + (−0.473 + 0.820i)17-s + (1.17 − 0.678i)19-s + (−1.61 − 0.930i)23-s + (−0.252 − 0.437i)25-s + (−1.11 − 0.643i)29-s + 0.633i·31-s + (−0.747 − 1.29i)37-s + (0.165 + 0.287i)41-s + (−0.881 + 1.52i)43-s − 0.258·47-s + (0.466 + 0.269i)53-s − 0.154i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.00584i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.999 + 0.00584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6490416578\)
\(L(\frac12)\) \(\approx\) \(0.6490416578\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 + 2.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.362 + 0.209i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.95 - 3.38i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.72 + 4.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.00 + 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.52iT - 31T^{2} \)
37 \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.06 - 1.84i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.77T + 47T^{2} \)
53 \( 1 + (-3.39 - 1.96i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.05T + 59T^{2} \)
61 \( 1 + 1.86iT - 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 + 8.51iT - 71T^{2} \)
73 \( 1 + (1.65 + 0.952i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 0.867T + 79T^{2} \)
83 \( 1 + (-3.45 + 5.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.88 + 8.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \)
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

−7.904500181410112464668461517506, −7.18034046098789144437141202403, −6.19562201933654940365793115803, −5.73330425280085774667963088017, −4.84361387924727460395773427622, −4.32282728465914202611481127268, −3.30633387510248596654984755336, −2.14937780761661227594042826440, −1.46045813170748020779091613306, −0.15566694410708169570411721236, 1.56211943514012981242934453640, 2.37824438816999149099236889316, 3.25118568386713662411713192819, 3.91489129951117150262139085618, 5.13133131770005898577392468883, 5.66427697453102668513103065561, 6.42503860010711884112507827373, 7.19790138721937686959001369447, 7.59143872622522703105858259848, 8.528539274061958881420771111858

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