Properties

Label 2-1155-385.384-c1-0-79
Degree $2$
Conductor $1155$
Sign $0.986 - 0.164i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 2.43·2-s + 3-s + 3.90·4-s + (2.21 − 0.290i)5-s + 2.43·6-s + (−1.43 + 2.22i)7-s + 4.64·8-s + 9-s + (5.38 − 0.705i)10-s + (−1.89 − 2.72i)11-s + 3.90·12-s + 2.23i·13-s + (−3.48 + 5.40i)14-s + (2.21 − 0.290i)15-s + 3.46·16-s + 4.51i·17-s + ⋯
L(s)  = 1  + 1.71·2-s + 0.577·3-s + 1.95·4-s + (0.991 − 0.129i)5-s + 0.992·6-s + (−0.542 + 0.840i)7-s + 1.64·8-s + 0.333·9-s + (1.70 − 0.223i)10-s + (−0.571 − 0.820i)11-s + 1.12·12-s + 0.619i·13-s + (−0.932 + 1.44i)14-s + (0.572 − 0.0749i)15-s + 0.866·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.986 - 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.688736040\)
\(L(\frac12)\) \(\approx\) \(5.688736040\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + (-2.21 + 0.290i)T \)
7 \( 1 + (1.43 - 2.22i)T \)
11 \( 1 + (1.89 + 2.72i)T \)
good2 \( 1 - 2.43T + 2T^{2} \)
13 \( 1 - 2.23iT - 13T^{2} \)
17 \( 1 - 4.51iT - 17T^{2} \)
19 \( 1 + 0.900T + 19T^{2} \)
23 \( 1 + 2.09iT - 23T^{2} \)
29 \( 1 + 3.99iT - 29T^{2} \)
31 \( 1 + 8.71iT - 31T^{2} \)
37 \( 1 + 7.32iT - 37T^{2} \)
41 \( 1 - 0.222T + 41T^{2} \)
43 \( 1 + 4.66T + 43T^{2} \)
47 \( 1 + 6.60T + 47T^{2} \)
53 \( 1 - 3.38iT - 53T^{2} \)
59 \( 1 - 8.40iT - 59T^{2} \)
61 \( 1 + 2.18T + 61T^{2} \)
67 \( 1 - 11.5iT - 67T^{2} \)
71 \( 1 - 2.71T + 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 - 4.44iT - 79T^{2} \)
83 \( 1 - 7.79iT - 83T^{2} \)
89 \( 1 - 1.90iT - 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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

−9.861775544301913888965672492255, −8.998985225617963338937809259530, −8.183054762570773342753407652429, −6.88765295693934311828476433140, −5.95694727832777512573269434009, −5.79206777806387927217094041889, −4.59408201289963976844583691706, −3.65062312509548947986006381854, −2.63790246086295563304278767616, −2.03791666097516345506314601061, 1.74165151801202190954322255718, 2.92651943234750570436064905249, 3.39685113933008418296562298309, 4.81702056338711389067850459909, 5.12803538704606587801916110140, 6.42941031651961958629985132254, 6.91255473424425595757598770755, 7.75358199500077892236233643489, 9.117823797081980820179636802634, 10.05196282769781238367327442923

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