↑

Properties

Label 2-2700-15.8-c1-0-23
Degree $2$
Conductor $2700$
Sign $-0.973 - 0.229i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
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  + (1.79 − 1.79i)7-s − 0.646i·11-s + (−3.79 − 3.79i)13-s + (−4.96 − 4.96i)17-s + 6.58i·19-s + (−1.87 + 1.87i)23-s − 7.99·29-s − 0.582·31-s + (−3 + 3i)37-s + 4.25i·41-s + (0.791 + 0.791i)43-s + (1.93 + 1.93i)47-s + 0.582i·49-s + (−5.47 + 5.47i)53-s − 12.8·59-s + ⋯
L(s)  = 1  + (0.677 − 0.677i)7-s − 0.194i·11-s + (−1.05 − 1.05i)13-s + (−1.20 − 1.20i)17-s + 1.51i·19-s + (−0.390 + 0.390i)23-s − 1.48·29-s − 0.104·31-s + (−0.493 + 0.493i)37-s + 0.664i·41-s + (0.120 + 0.120i)43-s + (0.282 + 0.282i)47-s + 0.0832i·49-s + (−0.752 + 0.752i)53-s − 1.67·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1558457944\)
\(L(\frac12)\) \(\approx\) \(0.1558457944\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \)
11 \( 1 + 0.646iT - 11T^{2} \)
13 \( 1 + (3.79 + 3.79i)T + 13iT^{2} \)
17 \( 1 + (4.96 + 4.96i)T + 17iT^{2} \)
19 \( 1 - 6.58iT - 19T^{2} \)
23 \( 1 + (1.87 - 1.87i)T - 23iT^{2} \)
29 \( 1 + 7.99T + 29T^{2} \)
31 \( 1 + 0.582T + 31T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 - 4.25iT - 41T^{2} \)
43 \( 1 + (-0.791 - 0.791i)T + 43iT^{2} \)
47 \( 1 + (-1.93 - 1.93i)T + 47iT^{2} \)
53 \( 1 + (5.47 - 5.47i)T - 53iT^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 - 6.16T + 61T^{2} \)
67 \( 1 + (-7 + 7i)T - 67iT^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-1.20 - 1.20i)T + 73iT^{2} \)
79 \( 1 + 6.16iT - 79T^{2} \)
83 \( 1 + (-1.22 + 1.22i)T - 83iT^{2} \)
89 \( 1 - 9.42T + 89T^{2} \)
97 \( 1 + (7.58 - 7.58i)T - 97iT^{2} \)
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

−8.214438745507657297060282399132, −7.68920134489268855796335723243, −7.12942874414371913195654955227, −6.05667952300360790108689412784, −5.23702046573544124443361755333, −4.54031532898446085563662013139, −3.61402540954691726899934425292, −2.58242177158613518127357205166, −1.47608738147509487381440740096, −0.04659057927145982793336365855, 1.95844297889775395239319350633, 2.28722100813272669889005517303, 3.78715873206343732224675429082, 4.62931197647700243084736442512, 5.21835777733531426113894089283, 6.26776376308000307711748440564, 6.96750765671747115776475706212, 7.69309804994885997189944499335, 8.723752147407684954269146971091, 9.031999984659882729573135360441

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