↑

Properties

Label 2-375-25.19-c1-0-14
Degree $2$
Conductor $375$
Sign $0.00690 + 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00690 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 375 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00690 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(375\)    =    \(3 \cdot 5^{3}\)
Sign: $0.00690 + 0.999i$
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: \((2,\ 375,\ (\ :1/2),\ 0.00690 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76070 - 1.74859i\)
\(L(\frac12)\) \(\approx\) \(1.76070 - 1.74859i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.01 + 0.655i)T + (1.61 - 1.17i)T^{2} \)
7 \( 1 + 4.35iT - 7T^{2} \)
11 \( 1 + (0.488 + 1.50i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.13 - 0.370i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.659 + 0.907i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-6.21 - 4.51i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-2.20 + 0.717i)T + (18.6 - 13.5i)T^{2} \)
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 \( 1 + (-6.06 - 1.96i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.30 - 7.10i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 1.24iT - 43T^{2} \)
47 \( 1 + (-2.42 - 3.33i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.20 + 3.03i)T + (-16.3 + 50.4i)T^{2} \)
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 \( 1 + (-2.27 + 3.12i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-8.57 + 6.23i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.75 - 1.54i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.7 + 8.55i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.13 - 7.06i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-3.10 - 9.54i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (4.40 + 6.06i)T + (-29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.24482803615004513889332734849, −10.78514178903264323514706384695, −9.634453346060760117152158061313, −7.999755008025773729989815031317, −7.15146329928248787486284337765, −6.08406408421091241840830559446, −5.14339063362152709580396726822, −4.02710893192116596558094249715, −3.16785116415968738438032988411, −1.31293693196435084519654718326, 2.58283997460390825301503695108, 3.69982183233393575866576721560, 5.09229680181108081319818006931, 5.44266527534409806169276635252, 6.41588031370308924515731886070, 7.54248231503844288848438812005, 8.992250491536385025814549466433, 9.624832979882697110851482732255, 11.14165125075305112388111413740, 11.82433405245017165282701005812

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