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Properties

Label 2-1100-11.5-c1-0-18
Degree $2$
Conductor $1100$
Sign $-0.780 + 0.625i$
Analytic cond. $8.78354$
Root an. cond. $2.96370$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.224i)3-s + (2.30 − 1.67i)7-s + (−0.881 − 2.71i)9-s + (−3.23 − 0.726i)11-s + (−1.42 − 4.39i)13-s + (−1.42 + 4.39i)17-s + (2.30 + 1.67i)19-s − 1.09·21-s − 6.47·23-s + (−0.690 + 2.12i)27-s + (−5.16 + 3.75i)29-s + (−1.80 − 5.56i)31-s + (0.836 + 0.951i)33-s + (3.92 − 2.85i)37-s + (−0.545 + 1.67i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.129i)3-s + (0.872 − 0.634i)7-s + (−0.293 − 0.904i)9-s + (−0.975 − 0.219i)11-s + (−0.395 − 1.21i)13-s + (−0.346 + 1.06i)17-s + (0.529 + 0.384i)19-s − 0.237·21-s − 1.34·23-s + (−0.132 + 0.409i)27-s + (−0.958 + 0.696i)29-s + (−0.324 − 0.999i)31-s + (0.145 + 0.165i)33-s + (0.645 − 0.469i)37-s + (−0.0872 + 0.268i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.780 + 0.625i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8704610755\)
\(L(\frac12)\) \(\approx\) \(0.8704610755\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
11 \( 1 + (3.23 + 0.726i)T \)
good3 \( 1 + (0.309 + 0.224i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.30 + 1.67i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.42 + 4.39i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.42 - 4.39i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.30 - 1.67i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (5.16 - 3.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.80 + 5.56i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-3.92 + 2.85i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (5.16 + 3.75i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-2.92 - 2.12i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.19 + 6.74i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.16 + 5.93i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.42 + 4.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.94T + 67T^{2} \)
71 \( 1 + (2.66 - 8.19i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (9.78 - 7.10i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.28 + 13.1i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.95 + 15.2i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (1.71 + 5.29i)T + (-78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.700636260912095556577858320169, −8.470798630661372509689226544745, −7.914381671558641706706616811873, −7.20179633510767075345059526419, −5.91075954587899780643151697672, −5.44496827103582770227737251493, −4.19427634329918853354415316167, −3.29313017320980551624107348335, −1.88579572256851502905315372517, −0.36901075647703763326187682799, 1.93450397623039838679152470309, 2.65509793435813778334697128252, 4.34101305678094069371093397719, 5.04655465397016711810382784161, 5.67672809741970352773882024415, 6.98235015189155539758551824478, 7.75488746217853196531101260362, 8.464026858974711516068468358132, 9.389527699540127873628009763519, 10.14830967545060741768227981919

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