↑

Properties

Label 2-10e2-20.3-c3-0-23
Degree $2$
Conductor $100$
Sign $-0.408 + 0.912i$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.75 − 2.21i)2-s + (6.14 − 6.14i)3-s + (−1.85 − 7.78i)4-s + (−2.86 − 24.4i)6-s + (16.4 + 16.4i)7-s + (−20.5 − 9.53i)8-s − 48.5i·9-s + 44.6i·11-s + (−59.2 − 36.4i)12-s + (−0.849 − 0.849i)13-s + (65.3 − 7.66i)14-s + (−57.1 + 28.8i)16-s + (−58.2 + 58.2i)17-s + (−107. − 85.1i)18-s + 23.7·19-s + ⋯
L(s)  = 1  + (0.619 − 0.784i)2-s + (1.18 − 1.18i)3-s + (−0.231 − 0.972i)4-s + (−0.194 − 1.66i)6-s + (0.888 + 0.888i)7-s + (−0.906 − 0.421i)8-s − 1.79i·9-s + 1.22i·11-s + (−1.42 − 0.877i)12-s + (−0.0181 − 0.0181i)13-s + (1.24 − 0.146i)14-s + (−0.892 + 0.450i)16-s + (−0.831 + 0.831i)17-s + (−1.41 − 1.11i)18-s + 0.286·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.408 + 0.912i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.408 + 0.912i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.62669 - 2.50979i\)
\(L(\frac12)\) \(\approx\) \(1.62669 - 2.50979i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.75 + 2.21i)T \)
5 \( 1 \)
good3 \( 1 + (-6.14 + 6.14i)T - 27iT^{2} \)
7 \( 1 + (-16.4 - 16.4i)T + 343iT^{2} \)
11 \( 1 - 44.6iT - 1.33e3T^{2} \)
13 \( 1 + (0.849 + 0.849i)T + 2.19e3iT^{2} \)
17 \( 1 + (58.2 - 58.2i)T - 4.91e3iT^{2} \)
19 \( 1 - 23.7T + 6.85e3T^{2} \)
23 \( 1 + (10.9 - 10.9i)T - 1.21e4iT^{2} \)
29 \( 1 + 127. iT - 2.43e4T^{2} \)
31 \( 1 + 253. iT - 2.97e4T^{2} \)
37 \( 1 + (-92.9 + 92.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 98.0T + 6.89e4T^{2} \)
43 \( 1 + (235. - 235. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-250. - 250. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-149. - 149. i)T + 1.48e5iT^{2} \)
59 \( 1 - 12.5T + 2.05e5T^{2} \)
61 \( 1 + 332.T + 2.26e5T^{2} \)
67 \( 1 + (-199. - 199. i)T + 3.00e5iT^{2} \)
71 \( 1 + 664. iT - 3.57e5T^{2} \)
73 \( 1 + (699. + 699. i)T + 3.89e5iT^{2} \)
79 \( 1 + 703.T + 4.93e5T^{2} \)
83 \( 1 + (940. - 940. i)T - 5.71e5iT^{2} \)
89 \( 1 - 386. iT - 7.04e5T^{2} \)
97 \( 1 + (-1.06e3 + 1.06e3i)T - 9.12e5iT^{2} \)
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

−12.99562835900649876664671701045, −12.29326431998502574746683908820, −11.33345356322122832124766711653, −9.680645510853069629600451944810, −8.671221583915651674816511723590, −7.54712913506507426071350904585, −6.07205899215595241874844273795, −4.35693115903214039206517385972, −2.51762745239399530776326585684, −1.71841528108085653284189000961, 3.05418449120907684557206904768, 4.17387842725484478183605730132, 5.18134442899648304466516483158, 7.12639530280360995537937433899, 8.333800575804676465806437637564, 8.956158837152879290063655190210, 10.45507719531725085586244963083, 11.51003876888888549159778284318, 13.36065869006557919796931093790, 14.05362815389747578818724598215

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