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Properties

Label 2-10e2-100.3-c1-0-7
Degree $2$
Conductor $100$
Sign $0.999 - 0.00124i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.40 − 0.132i)2-s + (−0.173 + 0.0274i)3-s + (1.96 − 0.373i)4-s + (−0.138 + 2.23i)5-s + (−0.240 + 0.0617i)6-s + (−2.95 − 2.95i)7-s + (2.71 − 0.785i)8-s + (−2.82 + 0.917i)9-s + (0.101 + 3.16i)10-s + (0.926 + 0.300i)11-s + (−0.330 + 0.118i)12-s + (1.59 − 3.12i)13-s + (−4.54 − 3.76i)14-s + (−0.0373 − 0.391i)15-s + (3.72 − 1.46i)16-s + (−0.216 + 1.36i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.0937i)2-s + (−0.100 + 0.0158i)3-s + (0.982 − 0.186i)4-s + (−0.0617 + 0.998i)5-s + (−0.0982 + 0.0251i)6-s + (−1.11 − 1.11i)7-s + (0.960 − 0.277i)8-s + (−0.941 + 0.305i)9-s + (0.0320 + 0.999i)10-s + (0.279 + 0.0907i)11-s + (−0.0954 + 0.0342i)12-s + (0.441 − 0.865i)13-s + (−1.21 − 1.00i)14-s + (−0.00965 − 0.101i)15-s + (0.930 − 0.366i)16-s + (−0.0524 + 0.330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.999 - 0.00124i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :1/2),\ 0.999 - 0.00124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51750 + 0.000942668i\)
\(L(\frac12)\) \(\approx\) \(1.51750 + 0.000942668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.132i)T \)
5 \( 1 + (0.138 - 2.23i)T \)
good3 \( 1 + (0.173 - 0.0274i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (2.95 + 2.95i)T + 7iT^{2} \)
11 \( 1 + (-0.926 - 0.300i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.59 + 3.12i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.216 - 1.36i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (5.71 - 4.14i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.463 - 0.910i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-4.15 + 5.72i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.87 - 6.71i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.27 - 1.16i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-1.03 - 3.19i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-4.54 + 4.54i)T - 43iT^{2} \)
47 \( 1 + (0.558 + 3.52i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.760 + 4.80i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-1.64 - 5.05i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.279 + 0.860i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.52 + 1.34i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-0.428 + 0.589i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.49 - 2.80i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (5.01 + 3.64i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.371 + 2.34i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-2.33 - 0.757i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.99 - 1.42i)T + (92.2 - 29.9i)T^{2} \)
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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

−13.84768819568199138886404183561, −13.05302005011394004340368298438, −11.85220426113492281681045410577, −10.60121451404853497244744823197, −10.26907360190145671294230745022, −8.017806028601768650497188709626, −6.68462168253564815446129137601, −5.98531018031683882546539223944, −4.00238435432411069257921429378, −2.92443056926737619227991100894, 2.71220618093969089087966621149, 4.39320130560484569766634422122, 5.80243458958650293458289539245, 6.55757730689612428845669689450, 8.534098537117711554329216134819, 9.312746602420865355764837052461, 11.18120229979611867870995262902, 12.08170759349587435996749972608, 12.76583041732259706201380971427, 13.71563970811095453079864986494

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