↑

Properties

Label 2-10e2-20.19-c10-0-85
Degree $2$
Conductor $100$
Sign $-0.584 - 0.811i$
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  + (27.3 − 16.5i)2-s − 17.1·3-s + (475. − 906. i)4-s + (−469. + 283. i)6-s + 2.88e3·7-s + (−1.98e3 − 3.27e4i)8-s − 5.87e4·9-s − 1.04e5i·11-s + (−8.14e3 + 1.55e4i)12-s + 1.45e5i·13-s + (7.89e4 − 4.77e4i)14-s + (−5.96e5 − 8.62e5i)16-s − 1.16e6i·17-s + (−1.60e6 + 9.72e5i)18-s + 4.52e6i·19-s + ⋯
L(s)  = 1  + (0.855 − 0.517i)2-s − 0.0704·3-s + (0.464 − 0.885i)4-s + (−0.0603 + 0.0364i)6-s + 0.171·7-s + (−0.0606 − 0.998i)8-s − 0.995·9-s − 0.651i·11-s + (−0.0327 + 0.0624i)12-s + 0.391i·13-s + (0.146 − 0.0887i)14-s + (−0.568 − 0.822i)16-s − 0.822i·17-s + (−0.851 + 0.514i)18-s + 1.82i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.584 - 0.811i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ -0.584 - 0.811i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.120770 + 0.235775i\)
\(L(\frac12)\) \(\approx\) \(0.120770 + 0.235775i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-27.3 + 16.5i)T \)
5 \( 1 \)
good3 \( 1 + 17.1T + 5.90e4T^{2} \)
7 \( 1 - 2.88e3T + 2.82e8T^{2} \)
11 \( 1 + 1.04e5iT - 2.59e10T^{2} \)
13 \( 1 - 1.45e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.16e6iT - 2.01e12T^{2} \)
19 \( 1 - 4.52e6iT - 6.13e12T^{2} \)
23 \( 1 + 3.50e6T + 4.14e13T^{2} \)
29 \( 1 + 2.68e7T + 4.20e14T^{2} \)
31 \( 1 + 2.04e7iT - 8.19e14T^{2} \)
37 \( 1 - 4.45e7iT - 4.80e15T^{2} \)
41 \( 1 - 2.94e7T + 1.34e16T^{2} \)
43 \( 1 + 1.50e8T + 2.16e16T^{2} \)
47 \( 1 - 1.88e8T + 5.25e16T^{2} \)
53 \( 1 - 5.81e8iT - 1.74e17T^{2} \)
59 \( 1 - 1.80e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.75e8T + 7.13e17T^{2} \)
67 \( 1 + 2.41e9T + 1.82e18T^{2} \)
71 \( 1 - 2.50e9iT - 3.25e18T^{2} \)
73 \( 1 - 2.32e8iT - 4.29e18T^{2} \)
79 \( 1 + 3.56e9iT - 9.46e18T^{2} \)
83 \( 1 + 3.50e9T + 1.55e19T^{2} \)
89 \( 1 - 9.00e9T + 3.11e19T^{2} \)
97 \( 1 - 4.72e9iT - 7.37e19T^{2} \)
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

−11.47462665501379173096098423678, −10.39331162363717432372060273525, −9.192500241872622690028958224005, −7.78365197710194868406737511880, −6.19568244986022487755923012899, −5.44115829520640676850957641197, −4.00835516807303661622978786149, −2.88715750211447691214077238260, −1.56099678366632581716059020078, −0.04367114136341775117960069134, 2.10581382061611825962839236605, 3.33991896230683792069657301451, 4.70679253730107166461762540319, 5.70462146382449500760642956169, 6.85950984783569232282074095465, 7.996608118713333453059135715769, 9.068952791109892758655213536000, 10.78042873534800170591171257363, 11.65432303814744644799096072220, 12.71379618222181117552449749353

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