↑

Properties

Label 2-1155-77.76-c1-0-20
Degree $2$
Conductor $1155$
Sign $0.206 + 0.978i$
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  − 2.19i·2-s + i·3-s − 2.83·4-s − i·5-s + 2.19·6-s + (−2.19 + 1.47i)7-s + 1.83i·8-s − 9-s − 2.19·10-s + (−1.23 + 3.07i)11-s − 2.83i·12-s + 6.52·13-s + (3.23 + 4.83i)14-s + 15-s − 1.63·16-s + 3.82·17-s + ⋯
L(s)  = 1  − 1.55i·2-s + 0.577i·3-s − 1.41·4-s − 0.447i·5-s + 0.897·6-s + (−0.830 + 0.556i)7-s + 0.647i·8-s − 0.333·9-s − 0.695·10-s + (−0.372 + 0.927i)11-s − 0.817i·12-s + 1.80·13-s + (0.864 + 1.29i)14-s + 0.258·15-s − 0.409·16-s + 0.927·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.206 + 0.978i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.206 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464615057\)
\(L(\frac12)\) \(\approx\) \(1.464615057\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 + iT \)
7 \( 1 + (2.19 - 1.47i)T \)
11 \( 1 + (1.23 - 3.07i)T \)
good2 \( 1 + 2.19iT - 2T^{2} \)
13 \( 1 - 6.52T + 13T^{2} \)
17 \( 1 - 3.82T + 17T^{2} \)
19 \( 1 - 2.33T + 19T^{2} \)
23 \( 1 - 3.83T + 23T^{2} \)
29 \( 1 - 4.90iT - 29T^{2} \)
31 \( 1 + 7.93iT - 31T^{2} \)
37 \( 1 + 7.93T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 6.37iT - 43T^{2} \)
47 \( 1 - 2.06iT - 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 3.76iT - 59T^{2} \)
61 \( 1 - 9.22T + 61T^{2} \)
67 \( 1 + 2.13T + 67T^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 - 8.34T + 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + 0.705iT - 89T^{2} \)
97 \( 1 - 10.0iT - 97T^{2} \)
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

−9.761266605621440080556341711832, −9.126689252518846482945867796062, −8.533071386976900247713713256636, −7.18871841438495714133301903144, −5.91883319350959626104043505545, −5.11089617919307119496368659117, −3.93253446747551201422524976864, −3.38364287695044529852111961592, −2.30515013466704664716181023011, −0.976315719235533066704641242546, 0.912845436834453462189460528480, 3.01950729485987480346561628870, 3.85087632003614123235142768544, 5.41511164278153919791234479485, 5.96918335008174337271985183621, 6.70058952536483087605603392148, 7.28099649547101144884003016570, 8.196766831185144695820273386422, 8.695887978721991578736685900006, 9.733536006463132668572849539929

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