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Properties

Label 2-55-55.43-c1-0-3
Degree $2$
Conductor $55$
Sign $-0.969 + 0.244i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.58 − 1.58i)2-s + (−1 − i)3-s + 3.00i·4-s + (−2 − i)5-s + 3.16i·6-s + (1.58 − 1.58i)8-s − i·9-s + (1.58 + 4.74i)10-s + (1 − 3.16i)11-s + (3.00 − 3.00i)12-s + (3.16 − 3.16i)13-s + (1 + 3i)15-s + 0.999·16-s + (3.16 + 3.16i)17-s + (−1.58 + 1.58i)18-s − 6.32·19-s + ⋯
L(s)  = 1  + (−1.11 − 1.11i)2-s + (−0.577 − 0.577i)3-s + 1.50i·4-s + (−0.894 − 0.447i)5-s + 1.29i·6-s + (0.559 − 0.559i)8-s − 0.333i·9-s + (0.500 + 1.50i)10-s + (0.301 − 0.953i)11-s + (0.866 − 0.866i)12-s + (0.877 − 0.877i)13-s + (0.258 + 0.774i)15-s + 0.249·16-s + (0.766 + 0.766i)17-s + (−0.372 + 0.372i)18-s − 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.969 + 0.244i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ -0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0433953 - 0.349167i\)
\(L(\frac12)\) \(\approx\) \(0.0433953 - 0.349167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
11 \( 1 + (-1 + 3.16i)T \)
good2 \( 1 + (1.58 + 1.58i)T + 2iT^{2} \)
3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 + (-3.16 - 3.16i)T + 17iT^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 - 6.32T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 6.32iT - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (-3 + 3i)T - 47iT^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + (-3 + 3i)T - 67iT^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.16 - 3.16i)T - 73iT^{2} \)
79 \( 1 - 6.32T + 79T^{2} \)
83 \( 1 + (-6.32 + 6.32i)T - 83iT^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (7 - 7i)T - 97iT^{2} \)
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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

−14.97974079313680884606138468962, −12.98485823157848293591959835471, −12.21118616335762390786025032360, −11.32660465527116186695832321215, −10.43076605168957514655098384351, −8.743308677963307139727859113879, −8.062227137583947885893163577620, −6.13754429632765401532898454305, −3.56769015484041802071715761194, −0.872296453380042686988010080295, 4.43129273790414915534528716769, 6.27960050060641700670605386254, 7.38974029877214882236176071903, 8.555156740161672823066773541136, 9.898819583401376212631158583627, 10.89944174647849326019021614739, 12.11982667470711951314687919039, 14.18921101149278700278016069382, 15.30452929556817541051919978519, 15.99415855402495081020900748480

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