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Properties

Label 2-55-11.2-c2-0-4
Degree $2$
Conductor $55$
Sign $0.126 + 0.992i$
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  + (−3.35 − 1.08i)2-s + (4.10 − 2.98i)3-s + (6.80 + 4.94i)4-s + (0.690 + 2.12i)5-s + (−16.9 + 5.52i)6-s + (4.13 − 5.68i)7-s + (−9.12 − 12.5i)8-s + (5.17 − 15.9i)9-s − 7.87i·10-s + (−10.8 + 1.62i)11-s + 42.6·12-s + (−1.31 − 0.427i)13-s + (−20.0 + 14.5i)14-s + (9.17 + 6.66i)15-s + (6.50 + 20.0i)16-s + (−8.29 + 2.69i)17-s + ⋯
L(s)  = 1  + (−1.67 − 0.544i)2-s + (1.36 − 0.994i)3-s + (1.70 + 1.23i)4-s + (0.138 + 0.425i)5-s + (−2.83 + 0.920i)6-s + (0.590 − 0.812i)7-s + (−1.14 − 1.57i)8-s + (0.575 − 1.77i)9-s − 0.787i·10-s + (−0.989 + 0.147i)11-s + 3.55·12-s + (−0.101 − 0.0328i)13-s + (−1.43 + 1.03i)14-s + (0.611 + 0.444i)15-s + (0.406 + 1.25i)16-s + (−0.487 + 0.158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.126 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.126 + 0.992i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ 0.126 + 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.641507 - 0.565119i\)
\(L(\frac12)\) \(\approx\) \(0.641507 - 0.565119i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.690 - 2.12i)T \)
11 \( 1 + (10.8 - 1.62i)T \)
good2 \( 1 + (3.35 + 1.08i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (-4.10 + 2.98i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-4.13 + 5.68i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (1.31 + 0.427i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (8.29 - 2.69i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-15.7 - 21.7i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 19.2T + 529T^{2} \)
29 \( 1 + (-7.84 + 10.7i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (13.6 - 41.9i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (35.4 + 25.7i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-27.4 - 37.7i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 0.643iT - 1.84e3T^{2} \)
47 \( 1 + (30.8 - 22.4i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-7.09 + 21.8i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-39.1 - 28.4i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-22.6 + 7.35i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 87.4T + 4.48e3T^{2} \)
71 \( 1 + (24.6 + 75.8i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (17.5 - 24.1i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (45.9 + 14.9i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (0.345 - 0.112i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 9.77T + 7.92e3T^{2} \)
97 \( 1 + (-13.2 + 40.7i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−14.70650922251690701886968265582, −13.66563437858386517669397979504, −12.44201684544231915207827331000, −10.95411378837580161888096272965, −9.944517584149485701047229002051, −8.669634676409086103674594659699, −7.74682627306232042725519048060, −7.11916481183521466628855324286, −3.00995630631733561918670153996, −1.56938921872291059891725622325, 2.45768593299320001043136398365, 5.15183504657077701184474577292, 7.46731824895412125341448367083, 8.576954721896496735735557940279, 9.059044490985148653331780557207, 10.07106011182628130508411869592, 11.24477540647909567639463730848, 13.42953815734827473598238283407, 14.87853370293468778073282034423, 15.53298017488779458875820959632

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