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Properties

Label 2-375-25.21-c1-0-4
Degree $2$
Conductor $375$
Sign $-0.485 - 0.874i$
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  + (−0.417 + 1.28i)2-s + (0.809 + 0.587i)3-s + (0.141 + 0.103i)4-s + (−1.09 + 0.793i)6-s + 1.59·7-s + (−2.37 + 1.72i)8-s + (0.309 + 0.951i)9-s + (1.02 − 3.16i)11-s + (0.0541 + 0.166i)12-s + (2.17 + 6.70i)13-s + (−0.666 + 2.05i)14-s + (−1.11 − 3.44i)16-s + (−3.31 + 2.40i)17-s − 1.35·18-s + (−0.459 + 0.333i)19-s + ⋯
L(s)  = 1  + (−0.295 + 0.908i)2-s + (0.467 + 0.339i)3-s + (0.0708 + 0.0515i)4-s + (−0.446 + 0.324i)6-s + 0.603·7-s + (−0.840 + 0.610i)8-s + (0.103 + 0.317i)9-s + (0.310 − 0.955i)11-s + (0.0156 + 0.0481i)12-s + (0.604 + 1.85i)13-s + (−0.178 + 0.547i)14-s + (−0.279 − 0.860i)16-s + (−0.803 + 0.583i)17-s − 0.318·18-s + (−0.105 + 0.0765i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.485 - 0.874i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.485 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.763719 + 1.29782i\)
\(L(\frac12)\) \(\approx\) \(0.763719 + 1.29782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.35258656860189301675442655249, −11.00377214238118206000419151552, −9.430165831376392109004647610268, −8.579811740523650537675094180362, −8.265785852439834265830911852350, −6.81798040980494056573636517850, −6.31414422557408511898525904119, −4.84875387097604100223842149026, −3.68163145468711899976468269196, −2.10864441787123680305267202953, 1.17998081097127194377119881608, 2.44379755000835032352440073552, 3.53821040595284605834180391328, 5.07483473369601417992662654571, 6.38513721448883925609433951013, 7.46095188698462755983249744667, 8.412158429398522916118106257789, 9.428199929813835176431319503866, 10.20635448093493278653461911702, 11.10878830598761653797392770903

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