Properties

Label 2-1100-5.4-c5-0-19
Degree $2$
Conductor $1100$
Sign $-0.447 - 0.894i$
Analytic cond. $176.422$
Root an. cond. $13.2824$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7i·3-s + 50i·7-s + 194·9-s + 121·11-s − 380i·13-s + 1.15e3i·17-s + 1.82e3·19-s − 350·21-s + 3.59e3i·23-s + 3.05e3i·27-s − 8.03e3·29-s − 2.94e3·31-s + 847i·33-s − 6.97e3i·37-s + 2.66e3·39-s + ⋯
L(s)  = 1  + 0.449i·3-s + 0.385i·7-s + 0.798·9-s + 0.301·11-s − 0.623i·13-s + 0.968i·17-s + 1.15·19-s − 0.173·21-s + 1.41i·23-s + 0.807i·27-s − 1.77·29-s − 0.550·31-s + 0.135i·33-s − 0.838i·37-s + 0.280·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(176.422\)
Root analytic conductor: \(13.2824\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1100} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1100,\ (\ :5/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.115057685\)
\(L(\frac12)\) \(\approx\) \(2.115057685\)
\(L(\frac{7}{2})\) 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 \( 1 - 121T \)
good3 \( 1 - 7iT - 243T^{2} \)
7 \( 1 - 50iT - 1.68e4T^{2} \)
13 \( 1 + 380iT - 3.71e5T^{2} \)
17 \( 1 - 1.15e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.82e3T + 2.47e6T^{2} \)
23 \( 1 - 3.59e3iT - 6.43e6T^{2} \)
29 \( 1 + 8.03e3T + 2.05e7T^{2} \)
31 \( 1 + 2.94e3T + 2.86e7T^{2} \)
37 \( 1 + 6.97e3iT - 6.93e7T^{2} \)
41 \( 1 + 520T + 1.15e8T^{2} \)
43 \( 1 + 2.48e3iT - 1.47e8T^{2} \)
47 \( 1 - 6.92e3iT - 2.29e8T^{2} \)
53 \( 1 + 1.37e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.17e4T + 7.14e8T^{2} \)
61 \( 1 - 3.41e4T + 8.44e8T^{2} \)
67 \( 1 - 6.15e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.49e4T + 1.80e9T^{2} \)
73 \( 1 + 3.64e4iT - 2.07e9T^{2} \)
79 \( 1 - 2.85e4T + 3.07e9T^{2} \)
83 \( 1 - 7.74e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.62e4T + 5.58e9T^{2} \)
97 \( 1 - 4.97e4iT - 8.58e9T^{2} \)
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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

−9.522555311178225235240628505380, −8.708344793092519895417178738007, −7.63125761581393574153560355830, −7.08754443631387839601495574807, −5.72518767860582419641585187946, −5.32765824077635548425516347533, −3.97351512405805675860523902094, −3.47842994481973804572668659036, −2.04793364656800932560014748981, −1.08434319231107670095343860710, 0.42611698236531113692811863463, 1.35808723449671520236525505996, 2.36835243554766557255993276927, 3.62472445142636645280562348991, 4.48033221431959045235411847088, 5.43262054418980756884997467348, 6.61232539777846507559541582022, 7.15273740425347729425782677851, 7.82610667234566693887726709428, 8.971761369428194800211218926765

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