Properties

Label 2-55-55.43-c1-0-1
Degree $2$
Conductor $55$
Sign $0.652 - 0.757i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.652 - 0.757i$
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: \((2,\ 55,\ (\ :1/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.999688 + 0.458388i\)
\(L(\frac12)\) \(\approx\) \(0.999688 + 0.458388i\)
\(L(\frac{3}{2})\) 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

−15.40221275747967212810364853272, −14.52671072538295847312890458669, −13.28479192337953652084431557195, −12.19377503205394523330612830766, −11.71904617684669320706421452753, −9.321113994849563190296681180799, −7.41604196646297359714148104156, −6.93234000252375324452646692331, −5.30018529982774542597949289403, −4.11659020628383720329043507720, 3.13099997286512682748889805950, 4.45905546606990116365670907887, 5.72639588141454045701669712650, 7.84273566115687712373307112076, 9.993905212072615523114276494335, 11.08179773469017204230085075962, 11.52091864690022716191318947871, 12.73787781447022779981680482504, 13.90717859542407994641636936900, 15.00518262449784763059434493755

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