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Properties

Label 2-10e2-20.19-c2-0-10
Degree $2$
Conductor $100$
Sign $0.887 - 0.460i$
Analytic cond. $2.72480$
Root an. cond. $1.65069$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.90 + 0.618i)2-s + 2.35·3-s + (3.23 + 2.35i)4-s + (4.47 + 1.45i)6-s − 5.25·7-s + (4.70 + 6.47i)8-s − 3.47·9-s − 19.9i·11-s + (7.60 + 5.52i)12-s + 8.47i·13-s + (−9.99 − 3.24i)14-s + (4.94 + 15.2i)16-s + 11.8i·17-s + (−6.60 − 2.14i)18-s − 15.2i·19-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + 0.783·3-s + (0.809 + 0.587i)4-s + (0.745 + 0.242i)6-s − 0.751·7-s + (0.587 + 0.809i)8-s − 0.385·9-s − 1.81i·11-s + (0.634 + 0.460i)12-s + 0.651i·13-s + (−0.714 − 0.232i)14-s + (0.309 + 0.951i)16-s + 0.699i·17-s + (−0.366 − 0.119i)18-s − 0.800i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.887 - 0.460i$
Analytic conductor: \(2.72480\)
Root analytic conductor: \(1.65069\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :1),\ 0.887 - 0.460i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.46648 + 0.602062i\)
\(L(\frac12)\) \(\approx\) \(2.46648 + 0.602062i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.90 - 0.618i)T \)
5 \( 1 \)
good3 \( 1 - 2.35T + 9T^{2} \)
7 \( 1 + 5.25T + 49T^{2} \)
11 \( 1 + 19.9iT - 121T^{2} \)
13 \( 1 - 8.47iT - 169T^{2} \)
17 \( 1 - 11.8iT - 289T^{2} \)
19 \( 1 + 15.2iT - 361T^{2} \)
23 \( 1 - 0.555T + 529T^{2} \)
29 \( 1 - 10.9T + 841T^{2} \)
31 \( 1 - 8.29iT - 961T^{2} \)
37 \( 1 + 18.3iT - 1.36e3T^{2} \)
41 \( 1 + 14.5T + 1.68e3T^{2} \)
43 \( 1 + 22.2T + 1.84e3T^{2} \)
47 \( 1 - 53.3T + 2.20e3T^{2} \)
53 \( 1 - 66.3iT - 2.80e3T^{2} \)
59 \( 1 - 17.4iT - 3.48e3T^{2} \)
61 \( 1 - 90.1T + 3.72e3T^{2} \)
67 \( 1 - 50.2T + 4.48e3T^{2} \)
71 \( 1 - 80.7iT - 5.04e3T^{2} \)
73 \( 1 - 5.55iT - 5.32e3T^{2} \)
79 \( 1 + 13.8iT - 6.24e3T^{2} \)
83 \( 1 - 76.2T + 6.88e3T^{2} \)
89 \( 1 - 111.T + 7.92e3T^{2} \)
97 \( 1 + 92.8iT - 9.40e3T^{2} \)
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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

−13.76839790015492874706690342591, −13.08110722411944202308560234063, −11.75475871796576736814675221567, −10.79310219858312558470979809218, −9.028195918966004735157581692346, −8.189462395226754127098019502804, −6.68533912294948568300827201414, −5.65540830760459105843990742828, −3.79160740727870800839734066070, −2.77754732514797996188992380849, 2.33515263780918919248888472837, 3.60772758216552286950204568921, 5.12491911948259606921823312430, 6.62534526615288142775058975503, 7.80530489977103349418472195830, 9.503412214161231106684494664313, 10.25012854889454910642036242508, 11.78920686957939300317682692797, 12.65987802520216440596248426833, 13.51026448278371766324801918689

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