Properties

Label 2-375-25.14-c1-0-2
Degree $2$
Conductor $375$
Sign $0.999 - 0.0237i$
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  + (−1.49 − 2.05i)2-s + (0.951 + 0.309i)3-s + (−1.37 + 4.22i)4-s + (−0.784 − 2.41i)6-s + 1.04i·7-s + (5.88 − 1.91i)8-s + (0.809 + 0.587i)9-s + (−2.40 + 1.74i)11-s + (−2.60 + 3.59i)12-s + (−3.33 + 4.58i)13-s + (2.13 − 1.55i)14-s + (−5.52 − 4.01i)16-s + (4.83 − 1.57i)17-s − 2.53i·18-s + (1.65 + 5.10i)19-s + ⋯
L(s)  = 1  + (−1.05 − 1.45i)2-s + (0.549 + 0.178i)3-s + (−0.685 + 2.11i)4-s + (−0.320 − 0.985i)6-s + 0.393i·7-s + (2.08 − 0.676i)8-s + (0.269 + 0.195i)9-s + (−0.724 + 0.526i)11-s + (−0.753 + 1.03i)12-s + (−0.924 + 1.27i)13-s + (0.570 − 0.414i)14-s + (−1.38 − 1.00i)16-s + (1.17 − 0.381i)17-s − 0.598i·18-s + (0.380 + 1.17i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.999 - 0.0237i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.999 - 0.0237i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.768930 + 0.00911895i\)
\(L(\frac12)\) \(\approx\) \(0.768930 + 0.00911895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.49 + 2.05i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 - 1.04iT - 7T^{2} \)
11 \( 1 + (2.40 - 1.74i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.33 - 4.58i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.83 + 1.57i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.65 - 5.10i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.26 - 3.12i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.210 - 0.646i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.262 - 0.808i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.950 - 1.30i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.942 + 0.684i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.68iT - 43T^{2} \)
47 \( 1 + (-3.12 - 1.01i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (12.0 + 3.92i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-2.59 - 1.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.38 + 3.18i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.883 + 0.287i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.436 - 1.34i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-6.65 - 9.16i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.447 + 1.37i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (10.9 - 3.54i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-7.33 + 5.33i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (5.73 + 1.86i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.32640289259761466441392022778, −10.14394735541055270126844762407, −9.751669264377028318749309562122, −8.968033818939266622010369383941, −7.964228176023380452262444191446, −7.23806803813759191513904535879, −5.19338832646056757525155421038, −3.83249869398142320089918877093, −2.73586559924613901378339890787, −1.69881970087476298852160564047, 0.71221150711561952310241657009, 2.95195659301426072230217259409, 4.92624511616611373519281541166, 5.80647875843382682277758170513, 7.01612050458390520480601891106, 7.74235445399069535520460382384, 8.271102156274137046993184213372, 9.327960870646483385373893501732, 10.11201379759006619019450808989, 10.83772571671999540133158307160

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