Properties

Label 2-10e2-25.4-c3-0-5
Degree $2$
Conductor $100$
Sign $-0.124 + 0.992i$
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  + (−4.13 + 5.68i)3-s + (−10.7 + 3.12i)5-s − 21.6i·7-s + (−6.91 − 21.2i)9-s + (3.15 − 9.72i)11-s + (72.0 − 23.4i)13-s + (26.5 − 73.9i)15-s + (−77.2 − 106. i)17-s + (−93.9 + 68.2i)19-s + (123. + 89.6i)21-s + (−85.3 − 27.7i)23-s + (105. − 67.0i)25-s + (−30.8 − 10.0i)27-s + (−96.2 − 69.9i)29-s + (−153. + 111. i)31-s + ⋯
L(s)  = 1  + (−0.794 + 1.09i)3-s + (−0.960 + 0.279i)5-s − 1.17i·7-s + (−0.256 − 0.788i)9-s + (0.0865 − 0.266i)11-s + (1.53 − 0.499i)13-s + (0.457 − 1.27i)15-s + (−1.10 − 1.51i)17-s + (−1.13 + 0.824i)19-s + (1.28 + 0.931i)21-s + (−0.774 − 0.251i)23-s + (0.843 − 0.536i)25-s + (−0.219 − 0.0714i)27-s + (−0.616 − 0.447i)29-s + (−0.889 + 0.645i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.124 + 0.992i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.256512 - 0.290708i\)
\(L(\frac12)\) \(\approx\) \(0.256512 - 0.290708i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (10.7 - 3.12i)T \)
good3 \( 1 + (4.13 - 5.68i)T + (-8.34 - 25.6i)T^{2} \)
7 \( 1 + 21.6iT - 343T^{2} \)
11 \( 1 + (-3.15 + 9.72i)T + (-1.07e3 - 782. i)T^{2} \)
13 \( 1 + (-72.0 + 23.4i)T + (1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (77.2 + 106. i)T + (-1.51e3 + 4.67e3i)T^{2} \)
19 \( 1 + (93.9 - 68.2i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + (85.3 + 27.7i)T + (9.84e3 + 7.15e3i)T^{2} \)
29 \( 1 + (96.2 + 69.9i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (153. - 111. i)T + (9.20e3 - 2.83e4i)T^{2} \)
37 \( 1 + (-207. + 67.5i)T + (4.09e4 - 2.97e4i)T^{2} \)
41 \( 1 + (17.4 + 53.6i)T + (-5.57e4 + 4.05e4i)T^{2} \)
43 \( 1 + 305. iT - 7.95e4T^{2} \)
47 \( 1 + (59.4 - 81.8i)T + (-3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (213. - 294. i)T + (-4.60e4 - 1.41e5i)T^{2} \)
59 \( 1 + (-57.9 - 178. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-48.2 + 148. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + (-264. - 364. i)T + (-9.29e4 + 2.86e5i)T^{2} \)
71 \( 1 + (133. + 97.1i)T + (1.10e5 + 3.40e5i)T^{2} \)
73 \( 1 + (921. + 299. i)T + (3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (450. + 327. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (732. + 1.00e3i)T + (-1.76e5 + 5.43e5i)T^{2} \)
89 \( 1 + (150. - 464. i)T + (-5.70e5 - 4.14e5i)T^{2} \)
97 \( 1 + (-611. + 842. i)T + (-2.82e5 - 8.68e5i)T^{2} \)
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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

−13.11968756723022332365430491154, −11.57680548105931320702151555668, −10.91372777182820165728590550161, −10.31857014262053601262921497292, −8.767572272755626673636854463551, −7.42529584537575472141860130426, −6.06740382049234195332687448670, −4.43289919821243900346294090904, −3.72079387455838664987806040031, −0.24360820494761716660252829016, 1.77070094641744348126638512344, 4.12571826877823871887721587684, 5.89992342711323120648866341492, 6.67784059018053572080900690031, 8.161100704463914371558841877427, 8.948730068512732817584810631914, 11.13458581754351828257968367047, 11.49487800800581507911784401806, 12.78612391041416466770853920877, 13.01096502727749369994535467481

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