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Properties

Label 2-55-5.4-c3-0-10
Degree $2$
Conductor $55$
Sign $-0.196 + 0.980i$
Analytic cond. $3.24510$
Root an. cond. $1.80141$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.54i·2-s + 4.97i·3-s − 4.53·4-s + (2.19 − 10.9i)5-s + 17.6·6-s − 25.4i·7-s − 12.2i·8-s + 2.25·9-s + (−38.8 − 7.78i)10-s + 11·11-s − 22.5i·12-s + 80.0i·13-s − 90.0·14-s + (54.5 + 10.9i)15-s − 79.7·16-s − 59.1i·17-s + ⋯
L(s)  = 1  − 1.25i·2-s + 0.957i·3-s − 0.567·4-s + (0.196 − 0.980i)5-s + 1.19·6-s − 1.37i·7-s − 0.541i·8-s + 0.0836·9-s + (−1.22 − 0.246i)10-s + 0.301·11-s − 0.543i·12-s + 1.70i·13-s − 1.71·14-s + (0.938 + 0.188i)15-s − 1.24·16-s − 0.844i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(3.24510\)
Root analytic conductor: \(1.80141\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :3/2),\ -0.196 + 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.944251 - 1.15237i\)
\(L(\frac12)\) \(\approx\) \(0.944251 - 1.15237i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.19 + 10.9i)T \)
11 \( 1 - 11T \)
good2 \( 1 + 3.54iT - 8T^{2} \)
3 \( 1 - 4.97iT - 27T^{2} \)
7 \( 1 + 25.4iT - 343T^{2} \)
13 \( 1 - 80.0iT - 2.19e3T^{2} \)
17 \( 1 + 59.1iT - 4.91e3T^{2} \)
19 \( 1 - 49.0T + 6.85e3T^{2} \)
23 \( 1 - 147. iT - 1.21e4T^{2} \)
29 \( 1 - 194.T + 2.43e4T^{2} \)
31 \( 1 + 162.T + 2.97e4T^{2} \)
37 \( 1 - 14.5iT - 5.06e4T^{2} \)
41 \( 1 - 213.T + 6.89e4T^{2} \)
43 \( 1 - 57.9iT - 7.95e4T^{2} \)
47 \( 1 - 415. iT - 1.03e5T^{2} \)
53 \( 1 + 453. iT - 1.48e5T^{2} \)
59 \( 1 - 300.T + 2.05e5T^{2} \)
61 \( 1 - 164.T + 2.26e5T^{2} \)
67 \( 1 - 44.6iT - 3.00e5T^{2} \)
71 \( 1 + 553.T + 3.57e5T^{2} \)
73 \( 1 + 589. iT - 3.89e5T^{2} \)
79 \( 1 + 356.T + 4.93e5T^{2} \)
83 \( 1 - 1.13e3iT - 5.71e5T^{2} \)
89 \( 1 - 963.T + 7.04e5T^{2} \)
97 \( 1 - 172. iT - 9.12e5T^{2} \)
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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

−14.10420166792883920709627908144, −13.27137685354397942329642879818, −11.91863963219621564106021756469, −11.01918024076974446322310146670, −9.760232003893692089045603103479, −9.321567604342897045589731961106, −7.10813584121527462013932960447, −4.64825197862189326414311462939, −3.82126084346972839481196400130, −1.30286162813950110678075769210, 2.51866876697626996885549066096, 5.63128293474074103981010727275, 6.44520629588537444998094133134, 7.58308121817611805411730405022, 8.586221766501351742048715103306, 10.43851077647007655091995682209, 11.97335522448937323731992315446, 13.01695306667889321629003822196, 14.38363767077700723136407811543, 15.12408045348956245948849067696

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