Properties

Label 2-375-25.4-c1-0-5
Degree $2$
Conductor $375$
Sign $0.999 + 0.0324i$
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  + (−0.234 − 0.0762i)2-s + (−0.587 + 0.809i)3-s + (−1.56 − 1.13i)4-s + (0.199 − 0.145i)6-s + 1.24i·7-s + (0.571 + 0.786i)8-s + (−0.309 − 0.951i)9-s + (0.794 − 2.44i)11-s + (1.84 − 0.599i)12-s + (4.45 − 1.44i)13-s + (0.0950 − 0.292i)14-s + (1.12 + 3.46i)16-s + (3.43 + 4.72i)17-s + 0.246i·18-s + (3.37 − 2.45i)19-s + ⋯
L(s)  = 1  + (−0.165 − 0.0539i)2-s + (−0.339 + 0.467i)3-s + (−0.784 − 0.569i)4-s + (0.0814 − 0.0592i)6-s + 0.471i·7-s + (0.201 + 0.278i)8-s + (−0.103 − 0.317i)9-s + (0.239 − 0.736i)11-s + (0.532 − 0.172i)12-s + (1.23 − 0.401i)13-s + (0.0254 − 0.0781i)14-s + (0.281 + 0.865i)16-s + (0.832 + 1.14i)17-s + 0.0581i·18-s + (0.773 − 0.562i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.999 + 0.0324i$
Analytic conductor: \(2.99439\)
Root analytic conductor: \(1.73043\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{375} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 375,\ (\ :1/2),\ 0.999 + 0.0324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00029 - 0.0162430i\)
\(L(\frac12)\) \(\approx\) \(1.00029 - 0.0162430i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.234 + 0.0762i)T + (1.61 + 1.17i)T^{2} \)
7 \( 1 - 1.24iT - 7T^{2} \)
11 \( 1 + (-0.794 + 2.44i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.45 + 1.44i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.43 - 4.72i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-3.37 + 2.45i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.52 - 0.496i)T + (18.6 + 13.5i)T^{2} \)
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 \( 1 + (1.21 - 0.394i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.68 - 8.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 3.88iT - 43T^{2} \)
47 \( 1 + (-1.88 + 2.59i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.83 - 10.7i)T + (-16.3 - 50.4i)T^{2} \)
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 \( 1 + (5.95 + 8.19i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (11.2 + 8.15i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (10.8 + 3.51i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.29 - 6.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.66 - 2.29i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-0.426 + 1.31i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (0.0179 - 0.0246i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.14650223694627795595811276709, −10.47661928481722032010100345907, −9.529023261201015385601999224934, −8.765463456503955386062977706521, −7.951246918443669599409503961558, −6.07331020243733424031214582003, −5.73368540541721657376946097864, −4.44616622486335651375232837641, −3.32121330379031862346577476743, −1.08456876211115104981969058674, 1.14353648887516584345484405783, 3.26196241704411418174011899102, 4.40477085466364127929820856563, 5.49999049852870900864093305395, 6.88580146069816768176774753861, 7.56570035254382100453929878365, 8.569315696664589096205182628570, 9.494613955372295578225376655730, 10.37788791350174516371934353902, 11.59672902258759724434629418121

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