Properties

Label 2-375-25.4-c1-0-7
Degree $2$
Conductor $375$
Sign $-0.0465 + 0.998i$
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  + (−1.28 − 0.417i)2-s + (0.587 − 0.809i)3-s + (−0.141 − 0.103i)4-s + (−1.09 + 0.793i)6-s + 1.59i·7-s + (1.72 + 2.37i)8-s + (−0.309 − 0.951i)9-s + (1.02 − 3.16i)11-s + (−0.166 + 0.0541i)12-s + (6.70 − 2.17i)13-s + (0.666 − 2.05i)14-s + (−1.11 − 3.44i)16-s + (−2.40 − 3.31i)17-s + 1.35i·18-s + (0.459 − 0.333i)19-s + ⋯
L(s)  = 1  + (−0.908 − 0.295i)2-s + (0.339 − 0.467i)3-s + (−0.0708 − 0.0515i)4-s + (−0.446 + 0.324i)6-s + 0.603i·7-s + (0.610 + 0.840i)8-s + (−0.103 − 0.317i)9-s + (0.310 − 0.955i)11-s + (−0.0481 + 0.0156i)12-s + (1.85 − 0.604i)13-s + (0.178 − 0.547i)14-s + (−0.279 − 0.860i)16-s + (−0.583 − 0.803i)17-s + 0.318i·18-s + (0.105 − 0.0765i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0465 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.0465 + 0.998i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.0465 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600902 - 0.629551i\)
\(L(\frac12)\) \(\approx\) \(0.600902 - 0.629551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (1.28 + 0.417i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.59iT - 7T^{2} \)
11 \( 1 + (-1.02 + 3.16i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-6.70 + 2.17i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.40 + 3.31i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.459 + 0.333i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.99 + 1.94i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.25 + 1.63i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.805 + 0.585i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.37 + 1.09i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.359 - 1.10i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.117iT - 43T^{2} \)
47 \( 1 + (-4.49 + 6.18i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.307 - 0.423i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.304 + 0.935i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.27 + 10.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.94 - 12.3i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-8.62 - 6.26i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (5.28 + 1.71i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.8 + 8.57i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.95 - 4.06i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (0.872 - 2.68i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.00 - 1.38i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.15009587745624499754686403576, −10.13795875612066916904061887829, −9.029370704124654786455943933887, −8.581444047900259641828657726813, −7.83420043015882731624540535015, −6.35370384670954733351962668467, −5.51505148926114651639615996853, −3.82595235364789763828143128252, −2.35771444067266078369250331864, −0.861441275304449815765104987040, 1.57547576429232580883070788701, 3.82329636537934269883985514311, 4.28019759413306721355544442729, 6.11664584049257017300348672786, 7.14130396239466857834748792843, 8.112889043500836560412598693481, 8.845593713872146037395372360053, 9.634718187459093563625498908857, 10.44581702466675609460962960290, 11.24189267872252588428636020301

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