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Properties

Label 2-375-25.16-c1-0-9
Degree $2$
Conductor $375$
Sign $-0.966 + 0.258i$
Analytic cond. $2.99439$
Root an. cond. $1.73043$
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  + (−1.63 − 1.18i)2-s + (0.309 + 0.951i)3-s + (0.641 + 1.97i)4-s + (0.623 − 1.92i)6-s − 1.01·7-s + (0.0473 − 0.145i)8-s + (−0.809 + 0.587i)9-s + (−3.85 − 2.79i)11-s + (−1.67 + 1.22i)12-s + (−0.0840 + 0.0610i)13-s + (1.66 + 1.20i)14-s + (3.10 − 2.25i)16-s + (1.80 − 5.55i)17-s + 2.01·18-s + (−0.223 + 0.688i)19-s + ⋯
L(s)  = 1  + (−1.15 − 0.839i)2-s + (0.178 + 0.549i)3-s + (0.320 + 0.987i)4-s + (0.254 − 0.783i)6-s − 0.385·7-s + (0.0167 − 0.0514i)8-s + (−0.269 + 0.195i)9-s + (−1.16 − 0.843i)11-s + (−0.484 + 0.352i)12-s + (−0.0232 + 0.0169i)13-s + (0.444 + 0.323i)14-s + (0.777 − 0.564i)16-s + (0.437 − 1.34i)17-s + 0.475·18-s + (−0.0513 + 0.158i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.966 + 0.258i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.966 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0394895 - 0.300611i\)
\(L(\frac12)\) \(\approx\) \(0.0394895 - 0.300611i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 \)
good2 \( 1 + (1.63 + 1.18i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + 1.01T + 7T^{2} \)
11 \( 1 + (3.85 + 2.79i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0840 - 0.0610i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.80 + 5.55i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.223 - 0.688i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (7.33 + 5.33i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.23 + 3.79i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.329 + 1.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.25 + 2.36i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.83 - 4.23i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.62T + 43T^{2} \)
47 \( 1 + (-2.53 - 7.79i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.34 - 4.15i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.97 + 2.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.63 + 4.09i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-3.06 + 9.43i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.33 + 10.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.98 - 5.07i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.767 - 2.36i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.31 - 4.03i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-14.8 - 10.8i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (2.07 + 6.37i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.73949568025955905715445446506, −9.970168164482893933322215699514, −9.434982886795274558381169177337, −8.307512990107021458540041149284, −7.79488670636659210323632841171, −6.11400067073454926093829246286, −4.94102130217043936795530537971, −3.31545567831214501352623679169, −2.39006180444384902046441239365, −0.27562050985500005229510225076, 1.81422051195478972686516118983, 3.59208319667484890899255887414, 5.42537669459715956286206734260, 6.41895860932814832386459844829, 7.35053399773555888518765795093, 8.007235289475603643145140588143, 8.755448537574346317684167801236, 10.00420127466812907932738414257, 10.26708401511650399102537982669, 11.82473084960083244992665735099

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