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Properties

Label 2-375-25.11-c1-0-12
Degree $2$
Conductor $375$
Sign $-0.0253 + 0.999i$
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  + (2.18 − 1.59i)2-s + (−0.309 + 0.951i)3-s + (1.64 − 5.06i)4-s + (0.836 + 2.57i)6-s − 0.470·7-s + (−2.78 − 8.55i)8-s + (−0.809 − 0.587i)9-s + (2.57 − 1.87i)11-s + (4.30 + 3.12i)12-s + (0.455 + 0.331i)13-s + (−1.02 + 0.748i)14-s + (−11.0 − 8.05i)16-s + (0.527 + 1.62i)17-s − 2.70·18-s + (1.15 + 3.55i)19-s + ⋯
L(s)  = 1  + (1.54 − 1.12i)2-s + (−0.178 + 0.549i)3-s + (0.822 − 2.53i)4-s + (0.341 + 1.05i)6-s − 0.177·7-s + (−0.982 − 3.02i)8-s + (−0.269 − 0.195i)9-s + (0.776 − 0.563i)11-s + (1.24 + 0.903i)12-s + (0.126 + 0.0918i)13-s + (−0.275 + 0.199i)14-s + (−2.77 − 2.01i)16-s + (0.127 + 0.393i)17-s − 0.637·18-s + (0.265 + 0.816i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0253 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0253 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.0253 + 0.999i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.0253 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99035 - 2.04138i\)
\(L(\frac12)\) \(\approx\) \(1.99035 - 2.04138i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 + 1.59i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + 0.470T + 7T^{2} \)
11 \( 1 + (-2.57 + 1.87i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.455 - 0.331i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.527 - 1.62i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.15 - 3.55i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.83 - 1.33i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.57 - 7.91i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.67 - 5.17i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.825 - 0.600i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.19 + 0.865i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 6.72T + 43T^{2} \)
47 \( 1 + (-1.37 + 4.21i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.17 + 6.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-10.4 - 7.56i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.102 - 0.0745i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.863 - 2.65i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.95 + 15.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.91 - 5.75i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.46 - 4.52i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.61 + 11.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.01 + 3.64i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.61 - 8.04i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.39535525643849888786171032666, −10.48054475229872883697463963625, −9.850362747209777510948000517926, −8.710283548580566255249681902890, −6.79184030021790933083171323388, −5.87192189336494354650999001919, −5.04196657838640854755302716699, −3.85187819064306511500516374268, −3.25944412760552519783532409088, −1.55124967005318200308572702618, 2.50045347786238596003330099131, 3.86668016565621147098537968395, 4.84323495675934446305743453916, 5.93491717488378090397774748745, 6.65897003635932453786576633330, 7.44606107328484488594320033227, 8.313326086412150276574717946952, 9.607768442999059232990570287078, 11.38710874896094050815405656615, 11.85102052286857075418652092760

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