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Properties

Label 2-375-25.9-c1-0-1
Degree $2$
Conductor $375$
Sign $-0.863 - 0.504i$
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  + (−1.59 + 2.18i)2-s + (−0.951 + 0.309i)3-s + (−1.64 − 5.06i)4-s + (0.836 − 2.57i)6-s − 0.470i·7-s + (8.55 + 2.78i)8-s + (0.809 − 0.587i)9-s + (2.57 + 1.87i)11-s + (3.12 + 4.30i)12-s + (−0.331 − 0.455i)13-s + (1.02 + 0.748i)14-s + (−11.0 + 8.05i)16-s + (1.62 + 0.527i)17-s + 2.70i·18-s + (−1.15 + 3.55i)19-s + ⋯
L(s)  = 1  + (−1.12 + 1.54i)2-s + (−0.549 + 0.178i)3-s + (−0.822 − 2.53i)4-s + (0.341 − 1.05i)6-s − 0.177i·7-s + (3.02 + 0.982i)8-s + (0.269 − 0.195i)9-s + (0.776 + 0.563i)11-s + (0.903 + 1.24i)12-s + (−0.0918 − 0.126i)13-s + (0.275 + 0.199i)14-s + (−2.77 + 2.01i)16-s + (0.393 + 0.127i)17-s + 0.637i·18-s + (−0.265 + 0.816i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.863 - 0.504i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.144366 + 0.533800i\)
\(L(\frac12)\) \(\approx\) \(0.144366 + 0.533800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 2.18i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 + 0.470iT - 7T^{2} \)
11 \( 1 + (-2.57 - 1.87i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.331 + 0.455i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.62 - 0.527i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.15 - 3.55i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.33 - 1.83i)T + (-7.10 - 21.8i)T^{2} \)
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 \( 1 + (-0.600 - 0.825i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.19 - 0.865i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 6.72iT - 43T^{2} \)
47 \( 1 + (4.21 - 1.37i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-6.70 + 2.17i)T + (42.8 - 31.1i)T^{2} \)
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 \( 1 + (-2.65 - 0.863i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.95 - 15.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.75 - 7.91i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.46 - 4.52i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-11.1 - 3.61i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (5.01 + 3.64i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-8.04 + 2.61i)T + (78.4 - 57.0i)T^{2} \)
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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

−11.45910103181043505772287390233, −10.34137532949807335375280578916, −9.818709517691396765078889434344, −8.898270991793515803353769389230, −7.923109263711541167890447228963, −7.05840501165120341843035807248, −6.25050090334095560708294427123, −5.37803455403514853977225006410, −4.24244502981518957752684568288, −1.30949109546315244269414199450, 0.67585808175589815924976301587, 2.15256285463753735065894854534, 3.47330260390285199274904022780, 4.67117059863506174985546993135, 6.38768040442430709794799616949, 7.57969545964445097231145969477, 8.591281535458492197588667991530, 9.297397641784557126160240151380, 10.27171725297273336740965900337, 10.94005615747013648693594181580

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