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Properties

Label 2-10e2-20.3-c3-0-5
Degree $2$
Conductor $100$
Sign $0.884 - 0.466i$
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  + (−1.52 − 2.38i)2-s + (−0.243 + 0.243i)3-s + (−3.34 + 7.26i)4-s + (0.952 + 0.208i)6-s + (−9.53 − 9.53i)7-s + (22.4 − 3.12i)8-s + 26.8i·9-s + 42.2i·11-s + (−0.956 − 2.58i)12-s + (41.7 + 41.7i)13-s + (−8.15 + 37.2i)14-s + (−41.6 − 48.5i)16-s + (34.1 − 34.1i)17-s + (64.0 − 41.0i)18-s + 130.·19-s + ⋯
L(s)  = 1  + (−0.539 − 0.841i)2-s + (−0.0468 + 0.0468i)3-s + (−0.417 + 0.908i)4-s + (0.0647 + 0.0141i)6-s + (−0.514 − 0.514i)7-s + (0.990 − 0.138i)8-s + 0.995i·9-s + 1.15i·11-s + (−0.0230 − 0.0622i)12-s + (0.889 + 0.889i)13-s + (−0.155 + 0.711i)14-s + (−0.650 − 0.759i)16-s + (0.487 − 0.487i)17-s + (0.838 − 0.537i)18-s + 1.57·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.884 - 0.466i$
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: \((2,\ 100,\ (\ :3/2),\ 0.884 - 0.466i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.935194 + 0.231563i\)
\(L(\frac12)\) \(\approx\) \(0.935194 + 0.231563i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.52 + 2.38i)T \)
5 \( 1 \)
good3 \( 1 + (0.243 - 0.243i)T - 27iT^{2} \)
7 \( 1 + (9.53 + 9.53i)T + 343iT^{2} \)
11 \( 1 - 42.2iT - 1.33e3T^{2} \)
13 \( 1 + (-41.7 - 41.7i)T + 2.19e3iT^{2} \)
17 \( 1 + (-34.1 + 34.1i)T - 4.91e3iT^{2} \)
19 \( 1 - 130.T + 6.85e3T^{2} \)
23 \( 1 + (107. - 107. i)T - 1.21e4iT^{2} \)
29 \( 1 + 80.4iT - 2.43e4T^{2} \)
31 \( 1 - 273. iT - 2.97e4T^{2} \)
37 \( 1 + (134. - 134. i)T - 5.06e4iT^{2} \)
41 \( 1 - 155.T + 6.89e4T^{2} \)
43 \( 1 + (223. - 223. i)T - 7.95e4iT^{2} \)
47 \( 1 + (72.1 + 72.1i)T + 1.03e5iT^{2} \)
53 \( 1 + (305. + 305. i)T + 1.48e5iT^{2} \)
59 \( 1 + 177.T + 2.05e5T^{2} \)
61 \( 1 + 41.4T + 2.26e5T^{2} \)
67 \( 1 + (-261. - 261. i)T + 3.00e5iT^{2} \)
71 \( 1 + 741. iT - 3.57e5T^{2} \)
73 \( 1 + (-516. - 516. i)T + 3.89e5iT^{2} \)
79 \( 1 - 222.T + 4.93e5T^{2} \)
83 \( 1 + (-824. + 824. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.44e3iT - 7.04e5T^{2} \)
97 \( 1 + (-196. + 196. i)T - 9.12e5iT^{2} \)
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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

−13.47060718939128872752623529238, −12.18485134071429936345704215138, −11.33049660722473285981009808787, −10.12460770692223054421854796052, −9.499631365015685697719757735838, −8.000205585186943765706065079347, −7.02384132619457870419587659567, −4.90127359285639769158026216712, −3.46503977062239871254055411718, −1.63489121613350363599985715943, 0.71734020551043217586601564123, 3.49694956282082995686245383868, 5.68711087722816403001818499772, 6.25270471975111154487658396830, 7.83338444664241953955428574124, 8.828340075472217346963374522171, 9.785792446845236863011458717868, 10.98075984643043841381054165612, 12.32256313257410372614102832727, 13.53149538218381306872244675320

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