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Properties

Label 2-375-25.16-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  + (−2.05 − 1.49i)2-s + (−0.309 − 0.951i)3-s + (1.37 + 4.22i)4-s + (−0.784 + 2.41i)6-s + 1.04·7-s + (1.91 − 5.88i)8-s + (−0.809 + 0.587i)9-s + (−2.40 − 1.74i)11-s + (3.59 − 2.60i)12-s + (−4.58 + 3.33i)13-s + (−2.13 − 1.55i)14-s + (−5.52 + 4.01i)16-s + (−1.57 + 4.83i)17-s + 2.53·18-s + (−1.65 + 5.10i)19-s + ⋯
L(s)  = 1  + (−1.45 − 1.05i)2-s + (−0.178 − 0.549i)3-s + (0.685 + 2.11i)4-s + (−0.320 + 0.985i)6-s + 0.393·7-s + (0.676 − 2.08i)8-s + (−0.269 + 0.195i)9-s + (−0.724 − 0.526i)11-s + (1.03 − 0.753i)12-s + (−1.27 + 0.924i)13-s + (−0.570 − 0.414i)14-s + (−1.38 + 1.00i)16-s + (−0.381 + 1.17i)17-s + 0.598·18-s + (−0.380 + 1.17i)19-s + ⋯

Functional equation

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

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: $\chi_{375} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.545 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191161 + 0.103627i\)
\(L(\frac12)\) \(\approx\) \(0.191161 + 0.103627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 1.49i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 - 1.04T + 7T^{2} \)
11 \( 1 + (2.40 + 1.74i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.58 - 3.33i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.57 - 4.83i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.65 - 5.10i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (3.12 + 2.26i)T + (7.10 + 21.8i)T^{2} \)
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 \( 1 + (-1.30 + 0.950i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.942 - 0.684i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 5.68T + 43T^{2} \)
47 \( 1 + (-1.01 - 3.12i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.92 - 12.0i)T + (-42.8 + 31.1i)T^{2} \)
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 \( 1 + (0.287 - 0.883i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.436 + 1.34i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (9.16 + 6.65i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.447 + 1.37i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.54 - 10.9i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.33 + 5.33i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.86 + 5.73i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.31169112609038733724414589464, −10.59057912345841963224664242580, −9.851428818421265572570236381973, −8.739870106452438083836608311015, −8.054335418532751666909074036320, −7.29778324240087929663031030392, −5.97939427229920043394538774415, −4.22760856073982870315110015246, −2.61149913793859011236677939908, −1.65280399366798223381898398187, 0.22853437360439165881431827692, 2.46597745265653408411328246823, 4.81152700994632800923314005390, 5.46336576297827101560239470181, 6.84786480743749253347557775347, 7.55903166555366877770840477405, 8.395067201675718698460447310584, 9.454513313754546548442675569314, 9.975730692882399513511519115911, 10.79969412140246139251032239601

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