↑

Properties

Label 2-2736-228.11-c1-0-18
Degree $2$
Conductor $2736$
Sign $0.995 + 0.0991i$
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  + (−0.420 − 0.242i)5-s + 1.92i·7-s − 0.610·11-s + (−0.889 − 1.54i)13-s + (3.89 + 2.25i)17-s + (−1.30 − 4.15i)19-s + (−0.121 − 0.209i)23-s + (−2.38 − 4.12i)25-s + (5.18 − 2.99i)29-s − 0.423i·31-s + (0.467 − 0.809i)35-s − 1.98·37-s + (5.42 + 3.13i)41-s + (3.24 + 1.87i)43-s + (2.88 + 4.99i)47-s + ⋯
L(s)  = 1  + (−0.187 − 0.108i)5-s + 0.728i·7-s − 0.184·11-s + (−0.246 − 0.427i)13-s + (0.945 + 0.546i)17-s + (−0.298 − 0.954i)19-s + (−0.0252 − 0.0437i)23-s + (−0.476 − 0.825i)25-s + (0.962 − 0.555i)29-s − 0.0759i·31-s + (0.0789 − 0.136i)35-s − 0.326·37-s + (0.847 + 0.489i)41-s + (0.495 + 0.286i)43-s + (0.420 + 0.728i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.995 + 0.0991i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.995 + 0.0991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.665265847\)
\(L(\frac12)\) \(\approx\) \(1.665265847\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 + 4.15i)T \)
good5 \( 1 + (0.420 + 0.242i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.92iT - 7T^{2} \)
11 \( 1 + 0.610T + 11T^{2} \)
13 \( 1 + (0.889 + 1.54i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.89 - 2.25i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.121 + 0.209i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.18 + 2.99i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.423iT - 31T^{2} \)
37 \( 1 + 1.98T + 37T^{2} \)
41 \( 1 + (-5.42 - 3.13i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.24 - 1.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.88 - 4.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.85 + 1.64i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.16 - 3.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.59 + 2.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.13 + 2.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.60 - 13.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.34 + 5.78i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.3 - 7.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.99T + 83T^{2} \)
89 \( 1 + (-8.14 + 4.70i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.18 + 5.52i)T + (-48.5 - 84.0i)T^{2} \)
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

−8.738380191441241866512483362943, −8.083682589059436324173941382633, −7.45636225328381609551248796533, −6.37696000852522048770834688311, −5.78844156476908211983791635284, −4.91816339532166440355202618587, −4.10904490627987034626698722365, −2.97729765462437799742851220274, −2.24637829439107452940323590582, −0.75981684221711074919932975588, 0.857152701288014421627809977921, 2.08611792019456957196030440712, 3.30628708592670714547335037585, 3.96860392018872945840472320035, 4.92909461703349305261510163682, 5.72641952395794977808379865731, 6.65487519331502746319228837046, 7.45706376441370355149725679435, 7.85390148707092426193682696694, 8.891623509860595528762010495584

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