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Properties

Label 2-55-11.2-c2-0-3
Degree $2$
Conductor $55$
Sign $0.353 - 0.935i$
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  + (2.92 + 0.951i)2-s + (−3.73 + 2.71i)3-s + (4.42 + 3.21i)4-s + (0.690 + 2.12i)5-s + (−13.5 + 4.39i)6-s + (6.38 − 8.78i)7-s + (2.66 + 3.66i)8-s + (3.80 − 11.7i)9-s + 6.88i·10-s + (10.3 + 3.66i)11-s − 25.2·12-s + (−10.8 − 3.52i)13-s + (27.0 − 19.6i)14-s + (−8.35 − 6.06i)15-s + (−2.45 − 7.55i)16-s + (−21.4 + 6.96i)17-s + ⋯
L(s)  = 1  + (1.46 + 0.475i)2-s + (−1.24 + 0.904i)3-s + (1.10 + 0.804i)4-s + (0.138 + 0.425i)5-s + (−2.25 + 0.732i)6-s + (0.911 − 1.25i)7-s + (0.332 + 0.458i)8-s + (0.423 − 1.30i)9-s + 0.688i·10-s + (0.942 + 0.333i)11-s − 2.10·12-s + (−0.834 − 0.271i)13-s + (1.93 − 1.40i)14-s + (−0.556 − 0.404i)15-s + (−0.153 − 0.472i)16-s + (−1.26 + 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.353 - 0.935i$
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: \((2,\ 55,\ (\ :1),\ 0.353 - 0.935i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.44260 + 0.996596i\)
\(L(\frac12)\) \(\approx\) \(1.44260 + 0.996596i\)
\(L(2)\) 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 \( 1 + (-10.3 - 3.66i)T \)
good2 \( 1 + (-2.92 - 0.951i)T + (3.23 + 2.35i)T^{2} \)
3 \( 1 + (3.73 - 2.71i)T + (2.78 - 8.55i)T^{2} \)
7 \( 1 + (-6.38 + 8.78i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (10.8 + 3.52i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (21.4 - 6.96i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-4.96 - 6.82i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 0.472T + 529T^{2} \)
29 \( 1 + (9.67 - 13.3i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-9.96 + 30.6i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (7.18 + 5.21i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-15.5 - 21.4i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 50.0iT - 1.84e3T^{2} \)
47 \( 1 + (28.3 - 20.6i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-10.3 + 31.9i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-89.4 - 65.0i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (71.4 - 23.2i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 - 44.4T + 4.48e3T^{2} \)
71 \( 1 + (16.2 + 49.9i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (46.3 - 63.7i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (121. + 39.3i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-35.6 + 11.5i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 108.T + 7.92e3T^{2} \)
97 \( 1 + (-19.1 + 58.7i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−14.99645376091290176612890343571, −14.44799688375634425589067357680, −13.18947100697681649172701774078, −11.78898188254862932368104120558, −11.04521197788173260052866296289, −9.877970409563187072439655448471, −7.27343686543474534188545626503, −6.15683293127585680532299158307, −4.77396529515831548768909366872, −4.08207161700978121689888146120, 2.03349421732793223270346297428, 4.75535116237023738824848257483, 5.59733902058064771381814299854, 6.75806087826619127982953720635, 8.839353634070666182908988373964, 11.17022490984381353349027799473, 11.82798420439188758911061038483, 12.32536182534878085856379522531, 13.44957559061749265356694245635, 14.52494789436209981167402688043

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