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Properties

Label 2-1100-55.54-c2-0-19
Degree $2$
Conductor $1100$
Sign $-0.0100 + 0.999i$
Analytic cond. $29.9728$
Root an. cond. $5.47474$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.65i·3-s − 9.09·7-s − 4.33·9-s + (9.88 − 4.82i)11-s + 22.5·13-s + 22.9·17-s + 16.0i·19-s + 33.1i·21-s − 0.141i·23-s − 17.0i·27-s + 29.8i·29-s + 21.5·31-s + (−17.6 − 36.1i)33-s − 28.0i·37-s − 82.1i·39-s + ⋯
L(s)  = 1  − 1.21i·3-s − 1.29·7-s − 0.481·9-s + (0.898 − 0.438i)11-s + 1.73·13-s + 1.34·17-s + 0.842i·19-s + 1.58i·21-s − 0.00616i·23-s − 0.631i·27-s + 1.02i·29-s + 0.694·31-s + (−0.533 − 1.09i)33-s − 0.757i·37-s − 2.10i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0100 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0100 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.0100 + 0.999i$
Analytic conductor: \(29.9728\)
Root analytic conductor: \(5.47474\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :1),\ -0.0100 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.974766307\)
\(L(\frac12)\) \(\approx\) \(1.974766307\)
\(L(2)\) 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 \( 1 + (-9.88 + 4.82i)T \)
good3 \( 1 + 3.65iT - 9T^{2} \)
7 \( 1 + 9.09T + 49T^{2} \)
13 \( 1 - 22.5T + 169T^{2} \)
17 \( 1 - 22.9T + 289T^{2} \)
19 \( 1 - 16.0iT - 361T^{2} \)
23 \( 1 + 0.141iT - 529T^{2} \)
29 \( 1 - 29.8iT - 841T^{2} \)
31 \( 1 - 21.5T + 961T^{2} \)
37 \( 1 + 28.0iT - 1.36e3T^{2} \)
41 \( 1 - 39.5iT - 1.68e3T^{2} \)
43 \( 1 + 21.4T + 1.84e3T^{2} \)
47 \( 1 + 15.4iT - 2.20e3T^{2} \)
53 \( 1 + 75.9iT - 2.80e3T^{2} \)
59 \( 1 - 50.4T + 3.48e3T^{2} \)
61 \( 1 + 98.4iT - 3.72e3T^{2} \)
67 \( 1 + 125. iT - 4.48e3T^{2} \)
71 \( 1 + 100.T + 5.04e3T^{2} \)
73 \( 1 - 128.T + 5.32e3T^{2} \)
79 \( 1 - 103. iT - 6.24e3T^{2} \)
83 \( 1 + 75.6T + 6.88e3T^{2} \)
89 \( 1 - 58.9T + 7.92e3T^{2} \)
97 \( 1 - 62.4iT - 9.40e3T^{2} \)
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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

−9.428791366997918275087113351298, −8.460975415264706737850119499736, −7.83219779810485024823690218139, −6.53888180854013213840627184088, −6.50905583037281705237978739701, −5.54070362041037201764175389849, −3.74993241454655377629209011927, −3.25701842154630828808623645560, −1.63067826997648992276158529962, −0.78567059187486852862633791641, 1.08483864845249685298671119703, 2.99023823426315891506598216323, 3.73962260380455557313889540306, 4.39373011425403957319798929372, 5.67776272831358057850337247297, 6.34619545332691697883759825636, 7.27091896178269907212085315401, 8.594323977414650028584114069710, 9.178026423146998236993398765576, 9.960895954653854932128653489269

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