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Properties

Label 2-375-25.19-c1-0-7
Degree $2$
Conductor $375$
Sign $0.707 - 0.706i$
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  + (−2.55 + 0.830i)2-s + (0.587 + 0.809i)3-s + (4.22 − 3.06i)4-s + (−2.17 − 1.57i)6-s − 1.68i·7-s + (−5.08 + 7.00i)8-s + (−0.309 + 0.951i)9-s + (0.333 + 1.02i)11-s + (4.96 + 1.61i)12-s + (2.54 + 0.827i)13-s + (1.40 + 4.31i)14-s + (3.95 − 12.1i)16-s + (2.31 − 3.18i)17-s − 2.68i·18-s + (−0.952 − 0.692i)19-s + ⋯
L(s)  = 1  + (−1.80 + 0.587i)2-s + (0.339 + 0.467i)3-s + (2.11 − 1.53i)4-s + (−0.887 − 0.644i)6-s − 0.637i·7-s + (−1.79 + 2.47i)8-s + (−0.103 + 0.317i)9-s + (0.100 + 0.309i)11-s + (1.43 + 0.465i)12-s + (0.706 + 0.229i)13-s + (0.374 + 1.15i)14-s + (0.989 − 3.04i)16-s + (0.560 − 0.771i)17-s − 0.633i·18-s + (−0.218 − 0.158i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.707 - 0.706i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.707 - 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.652069 + 0.269896i\)
\(L(\frac12)\) \(\approx\) \(0.652069 + 0.269896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good2 \( 1 + (2.55 - 0.830i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + 1.68iT - 7T^{2} \)
11 \( 1 + (-0.333 - 1.02i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-2.54 - 0.827i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.31 + 3.18i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.952 + 0.692i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.86 + 1.25i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.81 + 3.50i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.74 - 4.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-4.35 - 1.41i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.47 - 10.7i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2.58iT - 43T^{2} \)
47 \( 1 + (1.12 + 1.55i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.49 - 2.06i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.412 - 1.27i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.24 + 6.92i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (7.31 - 10.0i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-4.84 + 3.51i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.13 + 1.01i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (3.23 - 2.35i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.21 + 7.17i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.77 + 14.6i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.32 - 8.71i)T + (-29.9 + 92.2i)T^{2} \)
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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

−11.03323655589192722240601137659, −10.20866509075309062060406815638, −9.628211978864320788131379766013, −8.706696377451293699881697206026, −7.988360459507594606521859070933, −7.04511279259491833573793157697, −6.21350454119780311596877453181, −4.71370957332312060096804388588, −2.85592729735300475270001685442, −1.12528065074720057410490548778, 1.11509370468436591037681976836, 2.42407866591270622614186753026, 3.52405011327643326613527684958, 5.91477791413645084636384711589, 6.90166460488249234095836651352, 7.980598320543270040216322320727, 8.565324295826215820434304993485, 9.248874134849046351414663957933, 10.29434122200903667356470425839, 11.01604069869999243730290662526

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