↑

Properties

Label 2-1500-15.2-c1-0-24
Degree $2$
Conductor $1500$
Sign $-0.278 + 0.960i$
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  + (−1.51 + 0.835i)3-s + (1.05 + 1.05i)7-s + (1.60 − 2.53i)9-s − 5.36i·11-s + (2.50 − 2.50i)13-s + (−4.79 + 4.79i)17-s + 2.75i·19-s + (−2.48 − 0.719i)21-s + (−3.68 − 3.68i)23-s + (−0.316 + 5.18i)27-s − 3.14·29-s − 9.41·31-s + (4.48 + 8.13i)33-s + (3.00 + 3.00i)37-s + (−1.71 + 5.90i)39-s + ⋯
L(s)  = 1  + (−0.875 + 0.482i)3-s + (0.399 + 0.399i)7-s + (0.534 − 0.845i)9-s − 1.61i·11-s + (0.696 − 0.696i)13-s + (−1.16 + 1.16i)17-s + 0.631i·19-s + (−0.542 − 0.157i)21-s + (−0.769 − 0.769i)23-s + (−0.0608 + 0.998i)27-s − 0.583·29-s − 1.69·31-s + (0.780 + 1.41i)33-s + (0.494 + 0.494i)37-s + (−0.274 + 0.945i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.278 + 0.960i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.278 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6352681736\)
\(L(\frac12)\) \(\approx\) \(0.6352681736\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 0.835i)T \)
5 \( 1 \)
good7 \( 1 + (-1.05 - 1.05i)T + 7iT^{2} \)
11 \( 1 + 5.36iT - 11T^{2} \)
13 \( 1 + (-2.50 + 2.50i)T - 13iT^{2} \)
17 \( 1 + (4.79 - 4.79i)T - 17iT^{2} \)
19 \( 1 - 2.75iT - 19T^{2} \)
23 \( 1 + (3.68 + 3.68i)T + 23iT^{2} \)
29 \( 1 + 3.14T + 29T^{2} \)
31 \( 1 + 9.41T + 31T^{2} \)
37 \( 1 + (-3.00 - 3.00i)T + 37iT^{2} \)
41 \( 1 + 10.8iT - 41T^{2} \)
43 \( 1 + (1.30 - 1.30i)T - 43iT^{2} \)
47 \( 1 + (-3.22 + 3.22i)T - 47iT^{2} \)
53 \( 1 + (-0.347 - 0.347i)T + 53iT^{2} \)
59 \( 1 + 4.13T + 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 + (8.11 + 8.11i)T + 67iT^{2} \)
71 \( 1 + 6.87iT - 71T^{2} \)
73 \( 1 + (1.19 - 1.19i)T - 73iT^{2} \)
79 \( 1 + 7.55iT - 79T^{2} \)
83 \( 1 + (8.29 + 8.29i)T + 83iT^{2} \)
89 \( 1 - 17.1T + 89T^{2} \)
97 \( 1 + (-7.70 - 7.70i)T + 97iT^{2} \)
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

−9.054867110458619427510591430935, −8.636165214794698783429602974469, −7.75596951173086700451994713345, −6.42442786412224388719743297462, −5.91873807006959040820084331454, −5.31719025887142863606361138756, −4.07118455594928176446621272308, −3.45224950084926547054347713055, −1.82156402224460534492271712179, −0.28474664266726623524154487494, 1.44021323852115728649856149020, 2.32044387828349039089998064037, 4.13802253420938216500392642609, 4.66699220797634032973109634183, 5.59964678863381032644161266410, 6.65296307148556623234102960917, 7.21409670071065550417841570334, 7.77883543173225226330117975292, 9.112249118405331195796316950480, 9.657541210383774743422179727805

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