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Properties

Label 2-375-25.11-c1-0-10
Degree $2$
Conductor $375$
Sign $-0.0900 + 0.995i$
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  + (−0.881 + 0.640i)2-s + (−0.309 + 0.951i)3-s + (−0.251 + 0.772i)4-s + (−0.336 − 1.03i)6-s − 3.08·7-s + (−0.947 − 2.91i)8-s + (−0.809 − 0.587i)9-s + (0.929 − 0.674i)11-s + (−0.657 − 0.477i)12-s + (−3.30 − 2.39i)13-s + (2.72 − 1.97i)14-s + (1.38 + 1.00i)16-s + (1.42 + 4.40i)17-s + 1.08·18-s + (−1.84 − 5.67i)19-s + ⋯
L(s)  = 1  + (−0.623 + 0.452i)2-s + (−0.178 + 0.549i)3-s + (−0.125 + 0.386i)4-s + (−0.137 − 0.423i)6-s − 1.16·7-s + (−0.334 − 1.03i)8-s + (−0.269 − 0.195i)9-s + (0.280 − 0.203i)11-s + (−0.189 − 0.137i)12-s + (−0.915 − 0.665i)13-s + (0.727 − 0.528i)14-s + (0.346 + 0.252i)16-s + (0.346 + 1.06i)17-s + 0.256·18-s + (−0.423 − 1.30i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0900 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.0900 + 0.995i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ -0.0900 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0592764 - 0.0648757i\)
\(L(\frac12)\) \(\approx\) \(0.0592764 - 0.0648757i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.881 - 0.640i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 + (-0.929 + 0.674i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.30 + 2.39i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.42 - 4.40i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.84 + 5.67i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.88 + 1.36i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.63 - 5.02i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.182 + 0.560i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (9.22 + 6.70i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (7.67 + 5.57i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.42T + 43T^{2} \)
47 \( 1 + (-1.86 + 5.75i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.00 - 3.08i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.57 + 1.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (11.1 - 8.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-0.976 - 3.00i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.99 + 6.14i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (5.83 - 4.23i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.81 - 11.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.82 - 11.7i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.877 - 0.637i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.39 + 4.30i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.79001251878906118800476995541, −10.08348168433674971966154499791, −9.166153345378338843893877707615, −8.602577147148297206890888832975, −7.29182573143725096704159696938, −6.59648542360752355532153993528, −5.37797566308484199739680259288, −3.94005620783249047086771912496, −3.00185230774800760009624705569, −0.07159183286304459098940977879, 1.71579812843579168204134409825, 3.08673213315441955051401856688, 4.82762480443535882769627359704, 6.01002894577928346559113285386, 6.86858669307143576331535088686, 7.967405949805625817408811484077, 9.204554017949707037291336234776, 9.736085527196613755406653704765, 10.50051604501563835807742866931, 11.77491516292689814445755797154

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