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Properties

Label 2-55-11.10-c2-0-7
Degree $2$
Conductor $55$
Sign $-0.987 - 0.160i$
Analytic cond. $1.49864$
Root an. cond. $1.22419$
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  − 3.15i·2-s − 3.03·3-s − 5.98·4-s − 2.23·5-s + 9.57i·6-s − 5.67i·7-s + 6.26i·8-s + 0.187·9-s + 7.06i·10-s + (10.8 + 1.76i)11-s + 18.1·12-s − 17.2i·13-s − 17.9·14-s + 6.77·15-s − 4.13·16-s + 3.43i·17-s + ⋯
L(s)  = 1  − 1.57i·2-s − 1.01·3-s − 1.49·4-s − 0.447·5-s + 1.59i·6-s − 0.810i·7-s + 0.783i·8-s + 0.0208·9-s + 0.706i·10-s + (0.987 + 0.160i)11-s + 1.51·12-s − 1.32i·13-s − 1.27·14-s + 0.451·15-s − 0.258·16-s + 0.202i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.987 - 0.160i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ -0.987 - 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0524093 + 0.648606i\)
\(L(\frac12)\) \(\approx\) \(0.0524093 + 0.648606i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 2.23T \)
11 \( 1 + (-10.8 - 1.76i)T \)
good2 \( 1 + 3.15iT - 4T^{2} \)
3 \( 1 + 3.03T + 9T^{2} \)
7 \( 1 + 5.67iT - 49T^{2} \)
13 \( 1 + 17.2iT - 169T^{2} \)
17 \( 1 - 3.43iT - 289T^{2} \)
19 \( 1 + 18.5iT - 361T^{2} \)
23 \( 1 - 15.7T + 529T^{2} \)
29 \( 1 - 52.6iT - 841T^{2} \)
31 \( 1 - 27.5T + 961T^{2} \)
37 \( 1 + 40.1T + 1.36e3T^{2} \)
41 \( 1 + 74.8iT - 1.68e3T^{2} \)
43 \( 1 + 9.72iT - 1.84e3T^{2} \)
47 \( 1 - 18.2T + 2.20e3T^{2} \)
53 \( 1 + 75.2T + 2.80e3T^{2} \)
59 \( 1 - 1.75T + 3.48e3T^{2} \)
61 \( 1 + 48.5iT - 3.72e3T^{2} \)
67 \( 1 + 35.4T + 4.48e3T^{2} \)
71 \( 1 - 110.T + 5.04e3T^{2} \)
73 \( 1 - 50.7iT - 5.32e3T^{2} \)
79 \( 1 - 12.7iT - 6.24e3T^{2} \)
83 \( 1 + 112. iT - 6.88e3T^{2} \)
89 \( 1 - 66.3T + 7.92e3T^{2} \)
97 \( 1 - 65.1T + 9.40e3T^{2} \)
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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

−14.08998033146808571239278284980, −12.78387897470019522927957467807, −12.01967405071014108208869125902, −10.95538893274815877676355169457, −10.47565313283654609101839998342, −8.898168309271132704801085445848, −6.91474974857973089272406105728, −4.94518977783743698785110658354, −3.41922756034138270855533848218, −0.75953697274151190696236735053, 4.56537548038038301890701215711, 5.94978472258891478801654859894, 6.68508629409699528286438462545, 8.236092520293217724592927859845, 9.353872410422999849485830620812, 11.42397156280521576884081938410, 12.06093753548128649441764114182, 13.87677511339273139340924650512, 14.81097953249470112329216744191, 15.83203851648257569917798894645

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