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Properties

Label 2-1100-55.32-c1-0-12
Degree $2$
Conductor $1100$
Sign $0.850 + 0.525i$
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  + (1 − i)3-s + (3.31 − 3.31i)7-s + i·9-s + 3.31i·11-s + (3.31 + 3.31i)13-s + (3.31 − 3.31i)17-s − 6.63i·21-s + (−3 + 3i)23-s + (4 + 4i)27-s + 6.63·29-s − 4·31-s + (3.31 + 3.31i)33-s + (−5 − 5i)37-s + 6.63·39-s + (−3.31 − 3.31i)43-s + ⋯
L(s)  = 1  + (0.577 − 0.577i)3-s + (1.25 − 1.25i)7-s + 0.333i·9-s + 1.00i·11-s + (0.919 + 0.919i)13-s + (0.804 − 0.804i)17-s − 1.44i·21-s + (−0.625 + 0.625i)23-s + (0.769 + 0.769i)27-s + 1.23·29-s − 0.718·31-s + (0.577 + 0.577i)33-s + (−0.821 − 0.821i)37-s + 1.06·39-s + (−0.505 − 0.505i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.850 + 0.525i$
Analytic conductor: \(8.78354\)
Root analytic conductor: \(2.96370\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.388445798\)
\(L(\frac12)\) \(\approx\) \(2.388445798\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.31iT \)
good3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (-3.31 + 3.31i)T - 7iT^{2} \)
13 \( 1 + (-3.31 - 3.31i)T + 13iT^{2} \)
17 \( 1 + (-3.31 + 3.31i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (3 - 3i)T - 23iT^{2} \)
29 \( 1 - 6.63T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (3.31 + 3.31i)T + 43iT^{2} \)
47 \( 1 + (5 + 5i)T + 47iT^{2} \)
53 \( 1 + (-3 + 3i)T - 53iT^{2} \)
59 \( 1 - 10iT - 59T^{2} \)
61 \( 1 + 13.2iT - 61T^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-3.31 - 3.31i)T + 73iT^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (3.31 + 3.31i)T + 83iT^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 + (5 + 5i)T + 97iT^{2} \)
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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

−9.855661402684882943318612294296, −8.726787973936967013091159353555, −8.038452105911399376652920767911, −7.30584483625947897553012045024, −6.87859685570220996173127656087, −5.32748944776425961089887015161, −4.51340150887006159365340450850, −3.60673565925220604933252874432, −2.04978035851968256030555848026, −1.32950219168297337010734421521, 1.35207419724941572119890959015, 2.80441882781253588254289671506, 3.56998519319244199689659931933, 4.72215532312563727067919901051, 5.70689003166310928945395428664, 6.25703445908689658354984548570, 7.917098183019969423639500168121, 8.553102500030090488576620998693, 8.689339720058039493923642764189, 9.946850253992077072016399894104

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