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Properties

Label 2-375-25.9-c1-0-9
Degree $2$
Conductor $375$
Sign $0.727 + 0.686i$
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  + (−1.18 + 1.63i)2-s + (−0.951 + 0.309i)3-s + (−0.641 − 1.97i)4-s + (0.623 − 1.92i)6-s + 1.01i·7-s + (0.145 + 0.0473i)8-s + (0.809 − 0.587i)9-s + (−3.85 − 2.79i)11-s + (1.22 + 1.67i)12-s + (−0.0610 − 0.0840i)13-s + (−1.66 − 1.20i)14-s + (3.10 − 2.25i)16-s + (−5.55 − 1.80i)17-s + 2.01i·18-s + (0.223 − 0.688i)19-s + ⋯
L(s)  = 1  + (−0.839 + 1.15i)2-s + (−0.549 + 0.178i)3-s + (−0.320 − 0.987i)4-s + (0.254 − 0.783i)6-s + 0.385i·7-s + (0.0514 + 0.0167i)8-s + (0.269 − 0.195i)9-s + (−1.16 − 0.843i)11-s + (0.352 + 0.484i)12-s + (−0.0169 − 0.0232i)13-s + (−0.444 − 0.323i)14-s + (0.777 − 0.564i)16-s + (−1.34 − 0.437i)17-s + 0.475i·18-s + (0.0513 − 0.158i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.727 + 0.686i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.727 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.252341 - 0.100202i\)
\(L(\frac12)\) \(\approx\) \(0.252341 - 0.100202i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.18 - 1.63i)T + (-0.618 - 1.90i)T^{2} \)
7 \( 1 - 1.01iT - 7T^{2} \)
11 \( 1 + (3.85 + 2.79i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.0610 + 0.0840i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.55 + 1.80i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.223 + 0.688i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.33 + 7.33i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.23 - 3.79i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.329 + 1.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.36 + 3.25i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.83 - 4.23i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (-7.79 + 2.53i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.15 - 1.34i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.97 - 2.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.63 + 4.09i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (9.43 + 3.06i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.33 + 10.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.07 - 6.98i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.767 + 2.36i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.03 + 1.31i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (14.8 + 10.8i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.37 - 2.07i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.92756854121195404725841635688, −10.35041905755875248188665189786, −8.954992645165478555865843394589, −8.662941746216809107266430520438, −7.42226966086323276320856880444, −6.61463613891253140891778833883, −5.68733683901559771146985472860, −4.78635113291043436105805854200, −2.85121995166300503951295024470, −0.25467863306094881320491011988, 1.55459301911378759075924074682, 2.82151879284616758948032735818, 4.37331076279952494350129256135, 5.60961463586385428202632203079, 6.96746514460680234265370308121, 7.908221967336575215191219724495, 9.009621811458983279450459846372, 9.929246918714732206162023008541, 10.62797590960845793962148369415, 11.24659239742052160635632568753

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