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Properties

Label 2-375-25.6-c1-0-13
Degree $2$
Conductor $375$
Sign $-0.531 - 0.847i$
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  + (−0.655 − 2.01i)2-s + (0.809 − 0.587i)3-s + (−2.02 + 1.47i)4-s + (−1.71 − 1.24i)6-s − 4.35·7-s + (0.865 + 0.629i)8-s + (0.309 − 0.951i)9-s + (−0.488 − 1.50i)11-s + (−0.773 + 2.38i)12-s + (−0.370 + 1.13i)13-s + (2.85 + 8.79i)14-s + (−0.845 + 2.60i)16-s + (−0.907 − 0.659i)17-s − 2.12·18-s + (−6.21 − 4.51i)19-s + ⋯
L(s)  = 1  + (−0.463 − 1.42i)2-s + (0.467 − 0.339i)3-s + (−1.01 + 0.735i)4-s + (−0.700 − 0.509i)6-s − 1.64·7-s + (0.306 + 0.222i)8-s + (0.103 − 0.317i)9-s + (−0.147 − 0.453i)11-s + (−0.223 + 0.687i)12-s + (−0.102 + 0.315i)13-s + (0.763 + 2.35i)14-s + (−0.211 + 0.650i)16-s + (−0.220 − 0.159i)17-s − 0.500·18-s + (−1.42 − 1.03i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.531 - 0.847i$
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: \((2,\ 375,\ (\ :1/2),\ -0.531 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.246462 + 0.445664i\)
\(L(\frac12)\) \(\approx\) \(0.246462 + 0.445664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.655 + 2.01i)T + (-1.61 + 1.17i)T^{2} \)
7 \( 1 + 4.35T + 7T^{2} \)
11 \( 1 + (0.488 + 1.50i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.370 - 1.13i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.907 + 0.659i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (6.21 + 4.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.717 - 2.20i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.45 + 3.23i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.88 + 2.82i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.96 + 6.06i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.30 - 7.10i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.24T + 43T^{2} \)
47 \( 1 + (-3.33 + 2.42i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.03 + 2.20i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.82 + 8.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.431 - 1.32i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.12 + 2.27i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-8.57 + 6.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.54 + 4.75i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.7 - 8.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (7.06 + 5.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (3.10 + 9.54i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.06 - 4.40i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.76209156097740118027562020985, −9.811536782173023632297493558682, −9.228318549619461207243122382616, −8.447584679783850761852955627254, −6.96448659400918400215114448165, −6.14042066964215909001832118287, −4.15309340866185465365070616535, −3.12317580800354826047444026278, −2.27561653009304127252283521021, −0.35125395380730247462167170189, 2.78029299600114639528953453920, 4.14676490047641558036677464531, 5.56618514361311020401563506978, 6.50191857685542414863535514037, 7.16221127842335485615531706305, 8.343343969621273489996412536638, 8.956885587049423594995103768302, 9.927371957785606404494866247617, 10.50050214198868360746794286116, 12.29783683155626294100998107218

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