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Properties

Label 2-1100-5.4-c5-0-52
Degree $2$
Conductor $1100$
Sign $0.894 + 0.447i$
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  + 26.6i·3-s + 210. i·7-s − 466.·9-s + 121·11-s − 457. i·13-s + 531. i·17-s − 515.·19-s − 5.61e3·21-s + 2.84e3i·23-s − 5.94e3i·27-s + 443.·29-s − 3.97e3·31-s + 3.22e3i·33-s − 1.31e4i·37-s + 1.21e4·39-s + ⋯
L(s)  = 1  + 1.70i·3-s + 1.62i·7-s − 1.91·9-s + 0.301·11-s − 0.750i·13-s + 0.446i·17-s − 0.327·19-s − 2.77·21-s + 1.12i·23-s − 1.56i·27-s + 0.0978·29-s − 0.742·31-s + 0.515i·33-s − 1.58i·37-s + 1.28·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1100\)    =    \(2^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.894 + 0.447i$
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: \((2,\ 1100,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1280367804\)
\(L(\frac12)\) \(\approx\) \(0.1280367804\)
\(L(\frac{7}{2})\) 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 \( 1 - 26.6iT - 243T^{2} \)
7 \( 1 - 210. iT - 1.68e4T^{2} \)
13 \( 1 + 457. iT - 3.71e5T^{2} \)
17 \( 1 - 531. iT - 1.41e6T^{2} \)
19 \( 1 + 515.T + 2.47e6T^{2} \)
23 \( 1 - 2.84e3iT - 6.43e6T^{2} \)
29 \( 1 - 443.T + 2.05e7T^{2} \)
31 \( 1 + 3.97e3T + 2.86e7T^{2} \)
37 \( 1 + 1.31e4iT - 6.93e7T^{2} \)
41 \( 1 + 530.T + 1.15e8T^{2} \)
43 \( 1 + 1.91e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.33e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.93e4iT - 4.18e8T^{2} \)
59 \( 1 + 3.15e4T + 7.14e8T^{2} \)
61 \( 1 + 1.20e4T + 8.44e8T^{2} \)
67 \( 1 + 3.15e4iT - 1.35e9T^{2} \)
71 \( 1 + 9.16e3T + 1.80e9T^{2} \)
73 \( 1 + 2.69e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.54e4T + 3.07e9T^{2} \)
83 \( 1 + 4.71e4iT - 3.93e9T^{2} \)
89 \( 1 + 2.77e4T + 5.58e9T^{2} \)
97 \( 1 + 1.33e5iT - 8.58e9T^{2} \)
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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

−9.087096904233841052361777486352, −8.696333613990979224310694057391, −7.63356753734935064382727355613, −6.04809029971237095921209714073, −5.58554366271268116454335178780, −4.82905120711980314701645781314, −3.76201402131411278617520671787, −3.04670700992558287017569500125, −1.95640564894760578157892667047, −0.02650031426718303801556754171, 0.908241317144951174689116027231, 1.58119526631057442314114709293, 2.69570493109021607004450118214, 3.90336460812283686085735858350, 4.88497996730206777083167695478, 6.37980913655533982762595945666, 6.67803661739543867059495580651, 7.44815736038680089766635168503, 8.063473864004356841697414348360, 8.986214886033112834955706483253

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