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Properties

Label 2-55-11.6-c2-0-4
Degree $2$
Conductor $55$
Sign $0.994 - 0.108i$
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  + (1.57 − 0.512i)2-s + (2.41 + 1.75i)3-s + (−1.01 + 0.737i)4-s + (0.690 − 2.12i)5-s + (4.70 + 1.52i)6-s + (−1.31 − 1.80i)7-s + (−5.11 + 7.04i)8-s + (−0.0320 − 0.0986i)9-s − 3.70i·10-s + (−0.949 − 10.9i)11-s − 3.74·12-s + (−1.75 + 0.568i)13-s + (−2.98 − 2.17i)14-s + (5.39 − 3.92i)15-s + (−2.90 + 8.94i)16-s + (7.97 + 2.59i)17-s + ⋯
L(s)  = 1  + (0.787 − 0.256i)2-s + (0.804 + 0.584i)3-s + (−0.253 + 0.184i)4-s + (0.138 − 0.425i)5-s + (0.783 + 0.254i)6-s + (−0.187 − 0.257i)7-s + (−0.639 + 0.880i)8-s + (−0.00356 − 0.0109i)9-s − 0.370i·10-s + (−0.0862 − 0.996i)11-s − 0.311·12-s + (−0.134 + 0.0437i)13-s + (−0.213 − 0.155i)14-s + (0.359 − 0.261i)15-s + (−0.181 + 0.559i)16-s + (0.469 + 0.152i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(1.49864\)
Root analytic conductor: \(1.22419\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79124 + 0.0971539i\)
\(L(\frac12)\) \(\approx\) \(1.79124 + 0.0971539i\)
\(L(2)\) 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 \( 1 + (0.949 + 10.9i)T \)
good2 \( 1 + (-1.57 + 0.512i)T + (3.23 - 2.35i)T^{2} \)
3 \( 1 + (-2.41 - 1.75i)T + (2.78 + 8.55i)T^{2} \)
7 \( 1 + (1.31 + 1.80i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (1.75 - 0.568i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (-7.97 - 2.59i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (15.6 - 21.4i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 7.63T + 529T^{2} \)
29 \( 1 + (-31.8 - 43.7i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-1.08 - 3.32i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-37.8 + 27.5i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (15.9 - 21.9i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 14.9iT - 1.84e3T^{2} \)
47 \( 1 + (-58.0 - 42.1i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (12.1 + 37.4i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (29.1 - 21.1i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (44.0 + 14.3i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 60.8T + 4.48e3T^{2} \)
71 \( 1 + (37.8 - 116. i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (35.1 + 48.3i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (81.5 - 26.4i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (43.6 + 14.1i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 121.T + 7.92e3T^{2} \)
97 \( 1 + (10.2 + 31.6i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−14.61666715793757846189910194431, −14.10254890169820880273709418083, −12.97397489144528228532130332024, −11.99582292172056680324244034249, −10.38832094637274933134460001664, −9.016529936706516464057414141705, −8.213706890221770812603474180236, −5.87139879498671209113791105452, −4.26180830324559000391711888376, −3.14510052694506305719347208199, 2.64499136994518620688621738412, 4.58803548678198688954053132490, 6.25408955639800966496593171888, 7.53966781056248200707067952511, 9.043871941501991432257292568294, 10.23936504750582545333141117332, 12.10466849685415887816061351272, 13.17570266475333087106819317574, 13.86917624051539787446674601829, 14.86651484620473640732395204098

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