Properties

Label 2-2736-76.75-c1-0-40
Degree $2$
Conductor $2736$
Sign $-0.229 + 0.973i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + 2·5-s − 2.82i·7-s − 1.41i·11-s − 2.82i·13-s + 2·17-s + (1 − 4.24i)19-s − 1.41i·23-s − 25-s + 7.07i·29-s − 6·31-s − 5.65i·35-s − 11.3i·37-s + 4.24i·41-s − 7.07i·47-s − 1.00·49-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.06i·7-s − 0.426i·11-s − 0.784i·13-s + 0.485·17-s + (0.229 − 0.973i)19-s − 0.294i·23-s − 0.200·25-s + 1.31i·29-s − 1.07·31-s − 0.956i·35-s − 1.85i·37-s + 0.662i·41-s − 1.03i·47-s − 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.229 + 0.973i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.229 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.827789511\)
\(L(\frac12)\) \(\approx\) \(1.827789511\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (-1 + 4.24i)T \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 7.07iT - 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 - 4.24iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 7.07iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12.7iT - 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 - 2.82iT - 97T^{2} \)
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

−8.803836303869388025887237990287, −7.59024009090100770350262745389, −7.28228606117241987537414687676, −6.21323297088349995759190368928, −5.57221263482427776957652132679, −4.77119336982004184635187421274, −3.70746400731045660772215764563, −2.90984757051743393826411315754, −1.69550106553071754436707702664, −0.56960718636903515176045558222, 1.58969387208120941012034376764, 2.21803184637689717829514482767, 3.29888196616525037243434017917, 4.38894172675062966291398448840, 5.36368366270603245986997134831, 5.91695232788834883211118404340, 6.56653721126262848569220390314, 7.61204689926753097539005080221, 8.314135296785240751337418251325, 9.254953751879335105640281244228

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