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Properties

Label 2-375-25.14-c1-0-3
Degree $2$
Conductor $375$
Sign $-0.804 - 0.593i$
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  + (1.26 + 1.74i)2-s + (−0.951 − 0.309i)3-s + (−0.824 + 2.53i)4-s + (−0.667 − 2.05i)6-s + 3.16i·7-s + (−1.37 + 0.446i)8-s + (0.809 + 0.587i)9-s + (−1.24 + 0.904i)11-s + (1.56 − 2.15i)12-s + (−3.08 + 4.24i)13-s + (−5.52 + 4.01i)14-s + (1.79 + 1.30i)16-s + (1.22 − 0.398i)17-s + 2.16i·18-s + (−1.68 − 5.17i)19-s + ⋯
L(s)  = 1  + (0.897 + 1.23i)2-s + (−0.549 − 0.178i)3-s + (−0.412 + 1.26i)4-s + (−0.272 − 0.838i)6-s + 1.19i·7-s + (−0.485 + 0.157i)8-s + (0.269 + 0.195i)9-s + (−0.375 + 0.272i)11-s + (0.452 − 0.623i)12-s + (−0.854 + 1.17i)13-s + (−1.47 + 1.07i)14-s + (0.448 + 0.325i)16-s + (0.297 − 0.0967i)17-s + 0.509i·18-s + (−0.386 − 1.18i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.804 - 0.593i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.523042 + 1.59116i\)
\(L(\frac12)\) \(\approx\) \(0.523042 + 1.59116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 \)
good2 \( 1 + (-1.26 - 1.74i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 - 3.16iT - 7T^{2} \)
11 \( 1 + (1.24 - 0.904i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.08 - 4.24i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.22 + 0.398i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.68 + 5.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.78 - 5.21i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.730 + 2.24i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.37 + 4.24i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.50 + 4.81i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.90 - 5.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + (0.716 + 0.232i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.27 - 3.01i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.32 - 2.41i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.65 + 6.28i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.80 - 0.586i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.0219 - 0.0674i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.35 + 3.24i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.500 - 1.53i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (12.5 - 4.06i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.88 + 4.27i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (9.69 + 3.15i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.95581389940262961186397679463, −11.17785959738315383268011252923, −9.705086953109251582868760501763, −8.807000106085766605204993846801, −7.51535019365988861462199620147, −6.90672685787779683715613255247, −5.84018640124263973253154802170, −5.18271060031891798078213904140, −4.25506933759466102012291286980, −2.40148734057373357848822139455, 0.962904376754556537429635883759, 2.78685393634535111751169065380, 3.88203518605429771190240014430, 4.81604496247859460469439583087, 5.71857622272640182499339289166, 7.13363260571355897947547929459, 8.164176436685715878582127963226, 9.900556885680834709550424010602, 10.45766055885338855781820374981, 10.89735690219653033355850701980

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