Properties

Label 2-10e2-4.3-c6-0-28
Degree $2$
Conductor $100$
Sign $0.875 + 0.484i$
Analytic cond. $23.0054$
Root an. cond. $4.79639$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 7.74i)2-s − 30.9i·3-s + (−56.0 − 30.9i)4-s + (240. + 61.9i)6-s + 309. i·7-s + (352. − 371. i)8-s − 231.·9-s + 960. i·11-s + (−960. + 1.73e3i)12-s − 1.46e3·13-s + (−2.40e3 − 619. i)14-s + (2.17e3 + 3.47e3i)16-s + 4.76e3·17-s + (462. − 1.78e3i)18-s − 7.52e3i·19-s + ⋯
L(s)  = 1  + (−0.250 + 0.968i)2-s − 1.14i·3-s + (−0.875 − 0.484i)4-s + (1.11 + 0.286i)6-s + 0.903i·7-s + (0.687 − 0.726i)8-s − 0.316·9-s + 0.721i·11-s + (−0.555 + 1.00i)12-s − 0.667·13-s + (−0.874 − 0.225i)14-s + (0.531 + 0.847i)16-s + 0.970·17-s + (0.0792 − 0.306i)18-s − 1.09i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.875 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.875 + 0.484i$
Analytic conductor: \(23.0054\)
Root analytic conductor: \(4.79639\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3),\ 0.875 + 0.484i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.34242 - 0.346611i\)
\(L(\frac12)\) \(\approx\) \(1.34242 - 0.346611i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2 - 7.74i)T \)
5 \( 1 \)
good3 \( 1 + 30.9iT - 729T^{2} \)
7 \( 1 - 309. iT - 1.17e5T^{2} \)
11 \( 1 - 960. iT - 1.77e6T^{2} \)
13 \( 1 + 1.46e3T + 4.82e6T^{2} \)
17 \( 1 - 4.76e3T + 2.41e7T^{2} \)
19 \( 1 + 7.52e3iT - 4.70e7T^{2} \)
23 \( 1 + 1.04e4iT - 1.48e8T^{2} \)
29 \( 1 - 2.54e4T + 5.94e8T^{2} \)
31 \( 1 + 4.18e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.99e3T + 2.56e9T^{2} \)
41 \( 1 - 2.93e4T + 4.75e9T^{2} \)
43 \( 1 + 2.15e4iT - 6.32e9T^{2} \)
47 \( 1 + 7.56e3iT - 1.07e10T^{2} \)
53 \( 1 - 1.92e5T + 2.21e10T^{2} \)
59 \( 1 - 7.84e4iT - 4.21e10T^{2} \)
61 \( 1 + 1.09e4T + 5.15e10T^{2} \)
67 \( 1 + 3.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 5.32e5iT - 1.28e11T^{2} \)
73 \( 1 + 2.88e5T + 1.51e11T^{2} \)
79 \( 1 - 3.10e5iT - 2.43e11T^{2} \)
83 \( 1 + 2.04e5iT - 3.26e11T^{2} \)
89 \( 1 - 3.10e5T + 4.96e11T^{2} \)
97 \( 1 - 1.45e6T + 8.32e11T^{2} \)
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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

−12.69538817065064461822306441658, −11.98435732301681267878677398851, −10.14678712266239335333854943480, −9.043725963544900981115941502752, −7.88101053798562897690262562458, −7.05407740591684057546484934350, −6.01468497622942802714446379397, −4.71480608547640693120748589612, −2.28791116608141135436355143594, −0.64676256766681732266851722257, 1.11735431089136834325813856640, 3.21266746013634129799978007470, 4.11281181211473746291016240871, 5.32437986372162815565070486840, 7.50274177189736997003943230820, 8.760171459806232266670932038422, 10.08126348172967847569334761619, 10.27198648258874423863541295121, 11.48611020701202545169041995744, 12.57199759004697438714482682106

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