↑

Properties

Label 2-1500-15.2-c1-0-31
Degree $2$
Conductor $1500$
Sign $-0.940 - 0.339i$
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 − 1.62i)3-s + (−1.67 − 1.67i)7-s + (−2.30 − 1.91i)9-s − 3.38i·11-s + (−0.729 + 0.729i)13-s + (−4.39 + 4.39i)17-s − 5.19i·19-s + (−3.71 + 1.74i)21-s + (3.36 + 3.36i)23-s + (−4.47 + 2.63i)27-s − 6.24·29-s + 3.21·31-s + (−5.52 − 1.99i)33-s + (1.02 + 1.02i)37-s + (0.760 + 1.61i)39-s + ⋯
L(s)  = 1  + (0.339 − 0.940i)3-s + (−0.633 − 0.633i)7-s + (−0.769 − 0.638i)9-s − 1.02i·11-s + (−0.202 + 0.202i)13-s + (−1.06 + 1.06i)17-s − 1.19i·19-s + (−0.810 + 0.381i)21-s + (0.701 + 0.701i)23-s + (−0.861 + 0.507i)27-s − 1.15·29-s + 0.576·31-s + (−0.961 − 0.346i)33-s + (0.167 + 0.167i)37-s + (0.121 + 0.259i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.940 - 0.339i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6955567316\)
\(L(\frac12)\) \(\approx\) \(0.6955567316\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 1.62i)T \)
5 \( 1 \)
good7 \( 1 + (1.67 + 1.67i)T + 7iT^{2} \)
11 \( 1 + 3.38iT - 11T^{2} \)
13 \( 1 + (0.729 - 0.729i)T - 13iT^{2} \)
17 \( 1 + (4.39 - 4.39i)T - 17iT^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (-3.36 - 3.36i)T + 23iT^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 3.21T + 31T^{2} \)
37 \( 1 + (-1.02 - 1.02i)T + 37iT^{2} \)
41 \( 1 - 3.22iT - 41T^{2} \)
43 \( 1 + (1.41 - 1.41i)T - 43iT^{2} \)
47 \( 1 + (8.14 - 8.14i)T - 47iT^{2} \)
53 \( 1 + (1.42 + 1.42i)T + 53iT^{2} \)
59 \( 1 - 3.16T + 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 + (11.1 + 11.1i)T + 67iT^{2} \)
71 \( 1 + 16.4iT - 71T^{2} \)
73 \( 1 + (-9.57 + 9.57i)T - 73iT^{2} \)
79 \( 1 - 13.1iT - 79T^{2} \)
83 \( 1 + (-1.91 - 1.91i)T + 83iT^{2} \)
89 \( 1 - 3.69T + 89T^{2} \)
97 \( 1 + (1.81 + 1.81i)T + 97iT^{2} \)
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

−9.027931039168187822395705358752, −8.171711114203611485864407423814, −7.42053419703821069232825878796, −6.53666469546583763867122957802, −6.15533071625615641616126837201, −4.84038211969561843293390091880, −3.62521867198744544829087044539, −2.88548984134542435052033349822, −1.59993743391777051910975848993, −0.24735018190976352481344649590, 2.15028641430498112003733430284, 2.96663302899273622285612909397, 4.04204899055319368932146478726, 4.87092461422828048850894357635, 5.65842439697025685637955290779, 6.69673850082676993707333548517, 7.57194835888055157922751774493, 8.589064110790211951456610021126, 9.219422048545171610268163979389, 9.832996908780784912638286915539

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