Properties

Label 2-375-25.21-c1-0-5
Degree $2$
Conductor $375$
Sign $0.867 - 0.497i$
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  + (−0.0762 + 0.234i)2-s + (−0.809 − 0.587i)3-s + (1.56 + 1.13i)4-s + (0.199 − 0.145i)6-s + 1.24·7-s + (−0.786 + 0.571i)8-s + (0.309 + 0.951i)9-s + (0.794 − 2.44i)11-s + (−0.599 − 1.84i)12-s + (1.44 + 4.45i)13-s + (−0.0950 + 0.292i)14-s + (1.12 + 3.46i)16-s + (4.72 − 3.43i)17-s − 0.246·18-s + (−3.37 + 2.45i)19-s + ⋯
L(s)  = 1  + (−0.0539 + 0.165i)2-s + (−0.467 − 0.339i)3-s + (0.784 + 0.569i)4-s + (0.0814 − 0.0592i)6-s + 0.471·7-s + (−0.278 + 0.201i)8-s + (0.103 + 0.317i)9-s + (0.239 − 0.736i)11-s + (−0.172 − 0.532i)12-s + (0.401 + 1.23i)13-s + (−0.0254 + 0.0781i)14-s + (0.281 + 0.865i)16-s + (1.14 − 0.832i)17-s − 0.0581·18-s + (−0.773 + 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 - 0.497i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.867 - 0.497i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.867 - 0.497i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39671 + 0.372378i\)
\(L(\frac12)\) \(\approx\) \(1.39671 + 0.372378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0762 - 0.234i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 - 1.24T + 7T^{2} \)
11 \( 1 + (-0.794 + 2.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.44 - 4.45i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.72 + 3.43i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.37 - 2.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.496 - 1.52i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.60 - 1.89i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-7.43 + 5.40i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.394 - 1.21i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.68 - 8.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.88T + 43T^{2} \)
47 \( 1 + (2.59 + 1.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.7 + 7.83i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.97 + 6.07i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.18 + 3.63i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (8.19 - 5.95i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (11.2 + 8.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.51 + 10.8i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.29 + 6.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.29 - 1.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (0.426 - 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.0246 - 0.0179i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.60899832640630492164845641100, −10.87001214099340932922602497191, −9.628891235896032251684703807458, −8.364531239562678329729360530016, −7.73614313379594557862818965451, −6.59406649566409316923374995767, −5.99469992412124131614926375534, −4.53344791664948822636573996637, −3.14005959187795652608698013158, −1.60883599397637330483279363547, 1.27036380497927351049979602389, 2.87945960614785125193004372888, 4.42257046506458167393444002742, 5.57435838340046525509447229458, 6.34163417723688446383036119642, 7.47357712124270774677457229697, 8.523884281482220459227917530010, 9.882236978062407747404801544057, 10.46215821907458464377942681965, 11.08890530707479895730825018421

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