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Properties

Label 2-375-75.29-c2-0-1
Degree $2$
Conductor $375$
Sign $-0.475 + 0.879i$
Analytic cond. $10.2180$
Root an. cond. $3.19656$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.10 + 3.39i)2-s + (2.43 − 1.74i)3-s + (−7.07 − 5.14i)4-s + (3.23 + 10.2i)6-s − 0.132i·7-s + (13.7 − 9.96i)8-s + (2.90 − 8.51i)9-s + (−4.25 − 1.38i)11-s + (−26.2 − 0.182i)12-s + (−18.9 + 6.15i)13-s + (0.450 + 0.146i)14-s + (7.88 + 24.2i)16-s + (−18.9 + 13.7i)17-s + (25.7 + 19.2i)18-s + (−18.1 + 13.1i)19-s + ⋯
L(s)  = 1  + (−0.551 + 1.69i)2-s + (0.813 − 0.582i)3-s + (−1.76 − 1.28i)4-s + (0.539 + 1.70i)6-s − 0.0189i·7-s + (1.71 − 1.24i)8-s + (0.322 − 0.946i)9-s + (−0.387 − 0.125i)11-s + (−2.18 − 0.0152i)12-s + (−1.45 + 0.473i)13-s + (0.0321 + 0.0104i)14-s + (0.492 + 1.51i)16-s + (−1.11 + 0.809i)17-s + (1.42 + 1.06i)18-s + (−0.955 + 0.694i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.475 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $-0.475 + 0.879i$
Analytic conductor: \(10.2180\)
Root analytic conductor: \(3.19656\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1),\ -0.475 + 0.879i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.115131 - 0.193167i\)
\(L(\frac12)\) \(\approx\) \(0.115131 - 0.193167i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.43 + 1.74i)T \)
5 \( 1 \)
good2 \( 1 + (1.10 - 3.39i)T + (-3.23 - 2.35i)T^{2} \)
7 \( 1 + 0.132iT - 49T^{2} \)
11 \( 1 + (4.25 + 1.38i)T + (97.8 + 71.1i)T^{2} \)
13 \( 1 + (18.9 - 6.15i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (18.9 - 13.7i)T + (89.3 - 274. i)T^{2} \)
19 \( 1 + (18.1 - 13.1i)T + (111. - 343. i)T^{2} \)
23 \( 1 + (8.14 - 25.0i)T + (-427. - 310. i)T^{2} \)
29 \( 1 + (13.8 - 19.0i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-19.3 + 14.0i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (29.3 - 9.52i)T + (1.10e3 - 804. i)T^{2} \)
41 \( 1 + (31.0 - 10.0i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 52.1iT - 1.84e3T^{2} \)
47 \( 1 + (8.72 + 6.34i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-16.7 - 12.1i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (-56.7 + 18.4i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-26.7 + 82.2i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (14.7 + 20.3i)T + (-1.38e3 + 4.26e3i)T^{2} \)
71 \( 1 + (14.2 - 19.5i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-23.3 - 7.57i)T + (4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-5.10 - 3.70i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-0.0376 + 0.0273i)T + (2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + (42.9 + 13.9i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (75.4 - 103. i)T + (-2.90e3 - 8.94e3i)T^{2} \)
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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

−11.99427737810600848023046612765, −10.34998981056180539860155543060, −9.495712512733686932374922225923, −8.659507027886995211000762214087, −7.994477198272322003460899342293, −7.12545845143766566235308650375, −6.48773139277594790679061733308, −5.30813030002466476774747875355, −4.02344501817363914382576867702, −2.03738279946698858499720828414, 0.10100124155513230086874128122, 2.29029255389485501878787577377, 2.73459124074005836790516340011, 4.20514259999705389917006292308, 4.90309505210762351145293073260, 7.15143419982823840971787606560, 8.323402566784215341897145468781, 8.950641512555941272243070716528, 9.882126983961674425403138130486, 10.37506379554268926086059170164

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