Properties

Label 2-10e2-25.9-c3-0-7
Degree $2$
Conductor $100$
Sign $0.328 + 0.944i$
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  + (8.21 − 2.66i)3-s + (−9.50 − 5.89i)5-s − 24.1i·7-s + (38.5 − 27.9i)9-s + (−18.1 − 13.1i)11-s + (29.0 + 40.0i)13-s + (−93.7 − 23.0i)15-s + (66.2 + 21.5i)17-s + (−40.4 + 124. i)19-s + (−64.5 − 198. i)21-s + (114. − 157. i)23-s + (55.5 + 111. i)25-s + (104. − 143. i)27-s + (20.0 + 61.6i)29-s + (47.6 − 146. i)31-s + ⋯
L(s)  = 1  + (1.58 − 0.513i)3-s + (−0.849 − 0.526i)5-s − 1.30i·7-s + (1.42 − 1.03i)9-s + (−0.496 − 0.361i)11-s + (0.620 + 0.854i)13-s + (−1.61 − 0.396i)15-s + (0.944 + 0.306i)17-s + (−0.487 + 1.50i)19-s + (−0.670 − 2.06i)21-s + (1.03 − 1.42i)23-s + (0.444 + 0.895i)25-s + (0.745 − 1.02i)27-s + (0.128 + 0.394i)29-s + (0.276 − 0.849i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.80999 - 1.28621i\)
\(L(\frac12)\) \(\approx\) \(1.80999 - 1.28621i\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.50 + 5.89i)T \)
good3 \( 1 + (-8.21 + 2.66i)T + (21.8 - 15.8i)T^{2} \)
7 \( 1 + 24.1iT - 343T^{2} \)
11 \( 1 + (18.1 + 13.1i)T + (411. + 1.26e3i)T^{2} \)
13 \( 1 + (-29.0 - 40.0i)T + (-678. + 2.08e3i)T^{2} \)
17 \( 1 + (-66.2 - 21.5i)T + (3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (40.4 - 124. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + (-114. + 157. i)T + (-3.75e3 - 1.15e4i)T^{2} \)
29 \( 1 + (-20.0 - 61.6i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-47.6 + 146. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-43.7 - 60.2i)T + (-1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (180. - 131. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 - 401. iT - 7.95e4T^{2} \)
47 \( 1 + (257. - 83.7i)T + (8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (336. - 109. i)T + (1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (-557. + 404. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (168. + 122. i)T + (7.01e4 + 2.15e5i)T^{2} \)
67 \( 1 + (-587. - 190. i)T + (2.43e5 + 1.76e5i)T^{2} \)
71 \( 1 + (79.6 + 245. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-130. + 179. i)T + (-1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-246. - 757. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (961. + 312. i)T + (4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + (-398. - 289. i)T + (2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (1.32e3 - 431. i)T + (7.38e5 - 5.36e5i)T^{2} \)
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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

−13.24437314788010902842723792767, −12.53899112977051981642501386962, −11.01201141164565067339728410227, −9.731894036202814851609925485652, −8.316591813256023486083412592148, −7.994691393124456382846061212553, −6.75214898694507279151870626429, −4.29418916604880578609704202680, −3.32750521715915191699294119687, −1.25257919766425361913876719269, 2.62220899814560927453009634763, 3.45018167316930305515041608324, 5.16460329400046113758675439274, 7.22884209294174133898708294862, 8.283905839591279766621652039999, 9.005362085623754883968892659467, 10.18463384333976872798566391681, 11.43446479739271934066817760841, 12.71293447665351544099755666583, 13.74807392083746655118205758893

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