↑

Properties

Label 2-5292-9.7-c1-0-38
Degree $2$
Conductor $5292$
Sign $-0.999 + 0.0135i$
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  + (−0.764 − 1.32i)5-s + (0.417 − 0.723i)11-s + (1.81 + 3.13i)13-s + 0.602·17-s − 1.69·19-s + (−3.07 − 5.32i)23-s + (1.33 − 2.30i)25-s + (−4.99 + 8.65i)29-s + (1.65 + 2.86i)31-s − 8.79·37-s + (−3.51 − 6.09i)41-s + (0.846 − 1.46i)43-s + (4.23 − 7.33i)47-s − 7.99·53-s − 1.27·55-s + ⋯
L(s)  = 1  + (−0.341 − 0.592i)5-s + (0.125 − 0.218i)11-s + (0.502 + 0.870i)13-s + 0.146·17-s − 0.388·19-s + (−0.640 − 1.10i)23-s + (0.266 − 0.460i)25-s + (−0.927 + 1.60i)29-s + (0.297 + 0.514i)31-s − 1.44·37-s + (−0.549 − 0.951i)41-s + (0.129 − 0.223i)43-s + (0.617 − 1.06i)47-s − 1.09·53-s − 0.172·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.999 + 0.0135i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (3529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ -0.999 + 0.0135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2887386908\)
\(L(\frac12)\) \(\approx\) \(0.2887386908\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.764 + 1.32i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.417 + 0.723i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.81 - 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.602T + 17T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 + (3.07 + 5.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.99 - 8.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.65 - 2.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.79T + 37T^{2} \)
41 \( 1 + (3.51 + 6.09i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.846 + 1.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.23 + 7.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.99T + 53T^{2} \)
59 \( 1 + (-0.0652 - 0.112i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.38 + 4.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.12 - 1.94i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.39T + 71T^{2} \)
73 \( 1 - 4.50T + 73T^{2} \)
79 \( 1 + (7.87 - 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.16 + 5.47i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.06T + 89T^{2} \)
97 \( 1 + (7.76 - 13.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.011823281103621429144710420492, −6.90858747093495727246784901728, −6.63252852656429332909717394946, −5.54745595920379818901214177484, −4.92118600931641142693844482514, −4.05611094899829386194737844170, −3.49531569374007106403227374496, −2.25620824438630985327065651884, −1.35213590044492388050723244017, −0.07712865131482861209105199258, 1.36570881545843892917428823863, 2.44075048722716117051595565396, 3.39294705519142916596271616036, 3.91241420332338395413280379423, 4.92456563461231183934356827766, 5.79552318835013343141059297282, 6.32476250928463365976456756310, 7.24977013535223158814979533044, 7.81022042113981080487099815894, 8.338877499193313501901186079012

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