Properties

Label 2-10e2-20.7-c3-0-20
Degree $2$
Conductor $100$
Sign $-0.168 + 0.985i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.540 + 2.77i)2-s + (−2.23 − 2.23i)3-s + (−7.41 + 3.00i)4-s + (5 − 7.41i)6-s + (−11.1 + 11.1i)7-s + (−12.3 − 18.9i)8-s − 17i·9-s − 59.3i·11-s + (23.2 + 9.87i)12-s + (−26.5 + 26.5i)13-s + (−37.0 − 25.0i)14-s + (46.0 − 44.4i)16-s + (−79.5 − 79.5i)17-s + (47.1 − 9.18i)18-s − 59.3·19-s + ⋯
L(s)  = 1  + (0.191 + 0.981i)2-s + (−0.430 − 0.430i)3-s + (−0.927 + 0.375i)4-s + (0.340 − 0.504i)6-s + (−0.603 + 0.603i)7-s + (−0.545 − 0.838i)8-s − 0.629i·9-s − 1.62i·11-s + (0.560 + 0.237i)12-s + (−0.566 + 0.566i)13-s + (−0.707 − 0.477i)14-s + (0.718 − 0.695i)16-s + (−1.13 − 1.13i)17-s + (0.618 − 0.120i)18-s − 0.716·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.168 + 0.985i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.168 + 0.985i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.230663 - 0.273404i\)
\(L(\frac12)\) \(\approx\) \(0.230663 - 0.273404i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.540 - 2.77i)T \)
5 \( 1 \)
good3 \( 1 + (2.23 + 2.23i)T + 27iT^{2} \)
7 \( 1 + (11.1 - 11.1i)T - 343iT^{2} \)
11 \( 1 + 59.3iT - 1.33e3T^{2} \)
13 \( 1 + (26.5 - 26.5i)T - 2.19e3iT^{2} \)
17 \( 1 + (79.5 + 79.5i)T + 4.91e3iT^{2} \)
19 \( 1 + 59.3T + 6.85e3T^{2} \)
23 \( 1 + (-105. - 105. i)T + 1.21e4iT^{2} \)
29 \( 1 - 46iT - 2.43e4T^{2} \)
31 \( 1 - 118. iT - 2.97e4T^{2} \)
37 \( 1 + (212. + 212. i)T + 5.06e4iT^{2} \)
41 \( 1 + 188T + 6.89e4T^{2} \)
43 \( 1 + (122. + 122. i)T + 7.95e4iT^{2} \)
47 \( 1 + (194. - 194. i)T - 1.03e5iT^{2} \)
53 \( 1 + (26.5 - 26.5i)T - 1.48e5iT^{2} \)
59 \( 1 - 415.T + 2.05e5T^{2} \)
61 \( 1 - 72T + 2.26e5T^{2} \)
67 \( 1 + (-597. + 597. i)T - 3.00e5iT^{2} \)
71 \( 1 + 711. iT - 3.57e5T^{2} \)
73 \( 1 + (-504. + 504. i)T - 3.89e5iT^{2} \)
79 \( 1 + 830.T + 4.93e5T^{2} \)
83 \( 1 + (221. + 221. i)T + 5.71e5iT^{2} \)
89 \( 1 - 726iT - 7.04e5T^{2} \)
97 \( 1 + (-451. - 451. i)T + 9.12e5iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.22872555366892865566933997716, −12.26236627850845409164379553438, −11.22452540467941503332082417227, −9.333328688879723486473546411335, −8.733765716974204144601028819150, −7.05352933336555623736988167958, −6.31747300305657687987973364191, −5.18744952255073673429197293259, −3.33716251055838320621558203573, −0.19376129486343436405858270595, 2.21714214964251009182590597517, 4.12955091570625922833607507836, 5.01450385776734537300062592949, 6.76289508741175662181564752419, 8.434616900392715837207859805858, 10.02865390310006652712089619681, 10.29230996821072879064449518622, 11.44624775630450438583129582856, 12.80713680780432605217508854845, 13.14147924462653290303335087859

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