Properties

Label 2-10e2-4.3-c10-0-3
Degree $2$
Conductor $100$
Sign $-0.117 + 0.993i$
Analytic cond. $63.5357$
Root an. cond. $7.97093$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (21.2 + 23.9i)2-s + 94.4i·3-s + (−120. + 1.01e3i)4-s + (−2.25e3 + 2.00e3i)6-s + 2.00e4i·7-s + (−2.68e4 + 1.87e4i)8-s + 5.01e4·9-s − 1.98e5i·11-s + (−9.60e4 − 1.13e4i)12-s − 3.52e5·13-s + (−4.80e5 + 4.26e5i)14-s + (−1.01e6 − 2.45e5i)16-s − 1.97e6·17-s + (1.06e6 + 1.19e6i)18-s + 3.78e6i·19-s + ⋯
L(s)  = 1  + (0.664 + 0.747i)2-s + 0.388i·3-s + (−0.117 + 0.993i)4-s + (−0.290 + 0.258i)6-s + 1.19i·7-s + (−0.820 + 0.571i)8-s + 0.848·9-s − 1.23i·11-s + (−0.385 − 0.0457i)12-s − 0.950·13-s + (−0.893 + 0.793i)14-s + (−0.972 − 0.233i)16-s − 1.39·17-s + (0.563 + 0.634i)18-s + 1.52i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.117 + 0.993i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ -0.117 + 0.993i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.546951 - 0.615621i\)
\(L(\frac12)\) \(\approx\) \(0.546951 - 0.615621i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-21.2 - 23.9i)T \)
5 \( 1 \)
good3 \( 1 - 94.4iT - 5.90e4T^{2} \)
7 \( 1 - 2.00e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.98e5iT - 2.59e10T^{2} \)
13 \( 1 + 3.52e5T + 1.37e11T^{2} \)
17 \( 1 + 1.97e6T + 2.01e12T^{2} \)
19 \( 1 - 3.78e6iT - 6.13e12T^{2} \)
23 \( 1 + 4.41e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.04e7T + 4.20e14T^{2} \)
31 \( 1 - 1.45e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.27e8T + 4.80e15T^{2} \)
41 \( 1 - 9.37e7T + 1.34e16T^{2} \)
43 \( 1 + 2.01e8iT - 2.16e16T^{2} \)
47 \( 1 + 6.61e7iT - 5.25e16T^{2} \)
53 \( 1 + 8.60e7T + 1.74e17T^{2} \)
59 \( 1 + 5.09e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.18e8T + 7.13e17T^{2} \)
67 \( 1 + 7.20e8iT - 1.82e18T^{2} \)
71 \( 1 - 1.16e9iT - 3.25e18T^{2} \)
73 \( 1 + 1.07e9T + 4.29e18T^{2} \)
79 \( 1 + 3.63e9iT - 9.46e18T^{2} \)
83 \( 1 - 1.43e9iT - 1.55e19T^{2} \)
89 \( 1 - 9.94e8T + 3.11e19T^{2} \)
97 \( 1 + 4.59e9T + 7.37e19T^{2} \)
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

−12.60934942957412851981067187925, −11.97021644979255886848365741130, −10.55697483153208169469537082117, −9.092717989427982238736043136467, −8.313624097439965648200033820549, −6.89651045737842633093293352623, −5.80412282734409275469544825142, −4.79806332081830000569196898400, −3.55322692791306919012413132713, −2.21873858516466459998726099286, 0.15130672449608938696821145466, 1.41468283381311766470781332153, 2.50933464269373475800905016239, 4.22661413036285569166179713008, 4.78087852700724865427674065666, 6.76386434638567784119254237563, 7.29994598687820497960760782296, 9.351281816605269038280750324816, 10.18698071209357728544616937317, 11.15464712419460239925942548792

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