↑

Properties

Label 2-1500-25.19-c1-0-10
Degree $2$
Conductor $1500$
Sign $0.459 + 0.888i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (0.587 + 0.809i)3-s − 0.957i·7-s + (−0.309 + 0.951i)9-s + (−1.67 − 5.15i)11-s + (1.92 + 0.625i)13-s + (−0.377 + 0.520i)17-s + (−4.07 − 2.96i)19-s + (0.774 − 0.562i)21-s + (3.34 − 1.08i)23-s + (−0.951 + 0.309i)27-s + (8.20 − 5.96i)29-s + (−2.98 − 2.16i)31-s + (3.18 − 4.38i)33-s + (−10.7 − 3.49i)37-s + (0.625 + 1.92i)39-s + ⋯
L(s)  = 1  + (0.339 + 0.467i)3-s − 0.361i·7-s + (−0.103 + 0.317i)9-s + (−0.504 − 1.55i)11-s + (0.533 + 0.173i)13-s + (−0.0916 + 0.126i)17-s + (−0.935 − 0.679i)19-s + (0.169 − 0.122i)21-s + (0.697 − 0.226i)23-s + (−0.183 + 0.0594i)27-s + (1.52 − 1.10i)29-s + (−0.536 − 0.389i)31-s + (0.554 − 0.762i)33-s + (−1.76 − 0.574i)37-s + (0.100 + 0.308i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.459 + 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.557007369\)
\(L(\frac12)\) \(\approx\) \(1.557007369\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
good7 \( 1 + 0.957iT - 7T^{2} \)
11 \( 1 + (1.67 + 5.15i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-1.92 - 0.625i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.377 - 0.520i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (4.07 + 2.96i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-3.34 + 1.08i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-8.20 + 5.96i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.98 + 2.16i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (10.7 + 3.49i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.08 + 3.35i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 0.766iT - 43T^{2} \)
47 \( 1 + (2.90 + 3.99i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.49 - 4.81i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.45 - 4.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.34 - 4.13i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-5.59 + 7.70i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.66 + 7.02i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.16 - 1.67i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.58 + 6.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.819 + 1.12i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.527 - 1.62i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (8.57 + 11.8i)T + (-29.9 + 92.2i)T^{2} \)
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

−9.085742678776287795226174677894, −8.672734511002363522033144724889, −7.958951455236628411668318323943, −6.87826936376497824584447962463, −6.05204517435076102248556037198, −5.14898268849494776966630793527, −4.14169116664299210411572181890, −3.33837568826846780130518749703, −2.32852685133308146173881863896, −0.59858592694059947878027693382, 1.47376238176210304934600688903, 2.43544146825292850972120744235, 3.49797683769003897675075979412, 4.66941621184566958055094136358, 5.43837375132084244837732916266, 6.65928337122386743816922228654, 7.06768320809022997131464443987, 8.168190169485897164554937929111, 8.636581455619259290059070530387, 9.609343429738250419682657140293

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