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Properties

Label 2-375-25.14-c1-0-1
Degree $2$
Conductor $375$
Sign $0.545 - 0.837i$
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  + (−0.587 − 0.809i)2-s + (−0.951 − 0.309i)3-s + (0.309 − 0.951i)4-s + (0.309 + 0.951i)6-s + 4.47i·7-s + (−2.85 + 0.927i)8-s + (0.809 + 0.587i)9-s + (−2.61 + 1.90i)11-s + (−0.587 + 0.809i)12-s + (−1.98 + 2.73i)13-s + (3.61 − 2.62i)14-s + (0.809 + 0.587i)16-s + (2.71 − 0.881i)17-s − 0.999i·18-s + (1 + 3.07i)19-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (−0.549 − 0.178i)3-s + (0.154 − 0.475i)4-s + (0.126 + 0.388i)6-s + 1.69i·7-s + (−1.00 + 0.327i)8-s + (0.269 + 0.195i)9-s + (−0.789 + 0.573i)11-s + (−0.169 + 0.233i)12-s + (−0.551 + 0.758i)13-s + (0.966 − 0.702i)14-s + (0.202 + 0.146i)16-s + (0.658 − 0.213i)17-s − 0.235i·18-s + (0.229 + 0.706i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.545 - 0.837i$
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: \((2,\ 375,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522412 + 0.283159i\)
\(L(\frac12)\) \(\approx\) \(0.522412 + 0.283159i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.587 + 0.809i)T + (-0.618 + 1.90i)T^{2} \)
7 \( 1 - 4.47iT - 7T^{2} \)
11 \( 1 + (2.61 - 1.90i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.98 - 2.73i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.71 + 0.881i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1 - 3.07i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (2.62 + 3.61i)T + (-7.10 + 21.8i)T^{2} \)
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 \( 1 + (-4.75 + 6.54i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.11 - 0.812i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.70iT - 43T^{2} \)
47 \( 1 + (4.97 + 1.61i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.31 + 0.427i)T + (42.8 + 31.1i)T^{2} \)
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 \( 1 + (4.97 - 1.61i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.236 - 0.726i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.81 - 2.5i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.35 + 1.09i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.16 + 4.47i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-8.42 - 2.73i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.61374847302840844079290589518, −10.59065239295074197058361675750, −9.770978317860869257230018432718, −9.048477103782890711779759284068, −7.937772481672759944960536948542, −6.57895585163510688991286806922, −5.66332387734577003681600263277, −4.93433021605731431232315414194, −2.78058268904171661920998505644, −1.80438698724136430299997971061, 0.47264429947674406341020691398, 3.09686955299249117027453417222, 4.22157174662066579333343165409, 5.57005778641373710442005528743, 6.62573084880232433942628033092, 7.74671016390316078616283787867, 7.87840620523196485526255328837, 9.544411980282803006972516970772, 10.25659772750484654743801386694, 11.13782432564461519815680071548

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