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Properties

Label 2-375-25.11-c1-0-2
Degree $2$
Conductor $375$
Sign $-0.425 - 0.904i$
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  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (0.309 + 0.951i)6-s − 4.47·7-s + (0.927 + 2.85i)8-s + (−0.809 − 0.587i)9-s + (−2.61 + 1.90i)11-s + (−0.809 − 0.587i)12-s + (2.73 + 1.98i)13-s + (−3.61 + 2.62i)14-s + (0.809 + 0.587i)16-s + (0.881 + 2.71i)17-s − 0.999·18-s + (−1 − 3.07i)19-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (−0.154 + 0.475i)4-s + (0.126 + 0.388i)6-s − 1.69·7-s + (0.327 + 1.00i)8-s + (−0.269 − 0.195i)9-s + (−0.789 + 0.573i)11-s + (−0.233 − 0.169i)12-s + (0.758 + 0.551i)13-s + (−0.966 + 0.702i)14-s + (0.202 + 0.146i)16-s + (0.213 + 0.658i)17-s − 0.235·18-s + (−0.229 − 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.425 - 0.904i$
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: \((2,\ 375,\ (\ :1/2),\ -0.425 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.543497 + 0.856414i\)
\(L(\frac12)\) \(\approx\) \(0.543497 + 0.856414i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + 4.47T + 7T^{2} \)
11 \( 1 + (2.61 - 1.90i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.73 - 1.98i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.881 - 2.71i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1 + 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.61 - 2.62i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.35 + 4.16i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.23 - 6.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.54 - 4.75i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.11 - 0.812i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 + (-1.61 + 4.97i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.427 - 1.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.23 - 2.35i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.61 + 4.97i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.236 - 0.726i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.5 + 1.81i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.09 + 3.35i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (6.16 - 4.47i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.73 - 8.42i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.83058292750505546391653049986, −10.76952718477744667438909246308, −9.957995537037686192833493589495, −9.087697006882918872190762611178, −8.039800154780696598650359868363, −6.74450059422270821167302532168, −5.75762819200125097081027111976, −4.46787829613409200990459543090, −3.60608836262713645447694228499, −2.61239408439806242533916849068, 0.56630178829666160651123029828, 2.84525683551109382577733884676, 4.05969759060182111093038676770, 5.70710269430725567495492238378, 6.00843502614321834326984573606, 6.99268597979981460579800162028, 8.103092634282163162336186619527, 9.384946625230560669568031056521, 10.16650079905017294655202485691, 10.99353570035042862789235124171

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