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Properties

Label 2-55-55.43-c3-0-12
Degree $2$
Conductor $55$
Sign $-0.953 - 0.300i$
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  + (−2.85 − 2.85i)2-s + (3.30 + 3.30i)3-s + 8.35i·4-s + (−10.4 − 3.90i)5-s − 18.8i·6-s + (−15.7 − 15.7i)7-s + (1.01 − 1.01i)8-s − 5.21i·9-s + (18.7 + 41.1i)10-s + (−30.0 + 20.7i)11-s + (−27.5 + 27.5i)12-s + (−24.1 + 24.1i)13-s + 90.0i·14-s + (−21.6 − 47.4i)15-s + 61.0·16-s + (−53.3 − 53.3i)17-s + ⋯
L(s)  = 1  + (−1.01 − 1.01i)2-s + (0.635 + 0.635i)3-s + 1.04i·4-s + (−0.937 − 0.349i)5-s − 1.28i·6-s + (−0.850 − 0.850i)7-s + (0.0447 − 0.0447i)8-s − 0.192i·9-s + (0.594 + 1.30i)10-s + (−0.822 + 0.568i)11-s + (−0.663 + 0.663i)12-s + (−0.515 + 0.515i)13-s + 1.71i·14-s + (−0.373 − 0.817i)15-s + 0.953·16-s + (−0.760 − 0.760i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.953 - 0.300i$
Analytic conductor: \(3.24510\)
Root analytic conductor: \(1.80141\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :3/2),\ -0.953 - 0.300i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0402083 + 0.261149i\)
\(L(\frac12)\) \(\approx\) \(0.0402083 + 0.261149i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (10.4 + 3.90i)T \)
11 \( 1 + (30.0 - 20.7i)T \)
good2 \( 1 + (2.85 + 2.85i)T + 8iT^{2} \)
3 \( 1 + (-3.30 - 3.30i)T + 27iT^{2} \)
7 \( 1 + (15.7 + 15.7i)T + 343iT^{2} \)
13 \( 1 + (24.1 - 24.1i)T - 2.19e3iT^{2} \)
17 \( 1 + (53.3 + 53.3i)T + 4.91e3iT^{2} \)
19 \( 1 - 72.5T + 6.85e3T^{2} \)
23 \( 1 + (110. + 110. i)T + 1.21e4iT^{2} \)
29 \( 1 - 77.4T + 2.43e4T^{2} \)
31 \( 1 + 24.9T + 2.97e4T^{2} \)
37 \( 1 + (92.9 - 92.9i)T - 5.06e4iT^{2} \)
41 \( 1 - 148. iT - 6.89e4T^{2} \)
43 \( 1 + (-347. + 347. i)T - 7.95e4iT^{2} \)
47 \( 1 + (135. - 135. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-29.4 - 29.4i)T + 1.48e5iT^{2} \)
59 \( 1 + 350. iT - 2.05e5T^{2} \)
61 \( 1 + 462. iT - 2.26e5T^{2} \)
67 \( 1 + (663. - 663. i)T - 3.00e5iT^{2} \)
71 \( 1 + 388.T + 3.57e5T^{2} \)
73 \( 1 + (502. - 502. i)T - 3.89e5iT^{2} \)
79 \( 1 + 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + (263. - 263. i)T - 5.71e5iT^{2} \)
89 \( 1 - 206. iT - 7.04e5T^{2} \)
97 \( 1 + (-1.21e3 + 1.21e3i)T - 9.12e5iT^{2} \)
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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

−14.22319367305411237521878183000, −12.67051693915893846086903031017, −11.68275122175884865031516402611, −10.32819133095047878126217181662, −9.628860120026316265265093220054, −8.575649772235409178841452548606, −7.24646588319537075148890748634, −4.32689362915080084812110834620, −2.92306317014396494990781830554, −0.23480910711388243444766477596, 2.99584469448286624647947527285, 5.89053778142260344624925086604, 7.33336378441223599069385901587, 8.023776499317546971929510912275, 9.002313460527274177950403296027, 10.43888163090559881331434807163, 12.15193301668551745525541973955, 13.25216249113046796359241989434, 14.71720037319576587593427854583, 15.79595173078091983637329996254

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