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Properties

Label 2-375-25.6-c1-0-8
Degree $2$
Conductor $375$
Sign $0.662 - 0.749i$
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  + (0.474 + 1.46i)2-s + (0.809 − 0.587i)3-s + (−0.292 + 0.212i)4-s + (1.24 + 0.903i)6-s + 1.49·7-s + (2.03 + 1.48i)8-s + (0.309 − 0.951i)9-s + (−0.728 − 2.24i)11-s + (−0.111 + 0.343i)12-s + (0.417 − 1.28i)13-s + (0.710 + 2.18i)14-s + (−1.41 + 4.36i)16-s + (1.77 + 1.28i)17-s + 1.53·18-s + (−4.62 − 3.35i)19-s + ⋯
L(s)  = 1  + (0.335 + 1.03i)2-s + (0.467 − 0.339i)3-s + (−0.146 + 0.106i)4-s + (0.507 + 0.368i)6-s + 0.565·7-s + (0.720 + 0.523i)8-s + (0.103 − 0.317i)9-s + (−0.219 − 0.675i)11-s + (−0.0322 + 0.0991i)12-s + (0.115 − 0.355i)13-s + (0.189 + 0.584i)14-s + (−0.354 + 1.09i)16-s + (0.430 + 0.312i)17-s + 0.362·18-s + (−1.05 − 0.770i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92404 + 0.866900i\)
\(L(\frac12)\) \(\approx\) \(1.92404 + 0.866900i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.474 - 1.46i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.49T + 7T^{2} \)
11 \( 1 + (0.728 + 2.24i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-0.417 + 1.28i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.77 - 1.28i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.62 + 3.35i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.71 - 8.36i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (6.39 - 4.64i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.99 - 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.01 + 9.27i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.573 + 1.76i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.01T + 43T^{2} \)
47 \( 1 + (5.39 - 3.91i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.37 - 2.45i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.41 + 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.78 + 11.6i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.48 + 2.53i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (4.67 - 3.39i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.14 - 6.58i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.63 + 6.27i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.181 + 0.131i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.132 - 0.408i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (10.1 - 7.34i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.29629843958038682263531945238, −10.83276331464720530657730567436, −9.383646556466629133370675853653, −8.332799414961747175753561580596, −7.71715734160124153139256520362, −6.80037777653710622919193675221, −5.77544492413119631496847267708, −4.91156478845676162420988724249, −3.42225910588319856611063530969, −1.75532754403946982472098345084, 1.77123081400945059210876487986, 2.82235460623493918093462388783, 4.13411865676063491130938971419, 4.80704477143267448070026066414, 6.49941244406745317407850711894, 7.68652460781942774764842047459, 8.527833902470562700203449767794, 9.822726229548105217358539228790, 10.36010369323467774662197633641, 11.31733651228564825274325619580

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