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Properties

Label 2-55-55.32-c1-0-1
Degree $2$
Conductor $55$
Sign $0.278 - 0.960i$
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  + (−2.15 + 2.15i)3-s + 2i·4-s + (1.65 − 1.5i)5-s − 6.31i·9-s + 3.31·11-s + (−4.31 − 4.31i)12-s + (−0.341 + 6.81i)15-s − 4·16-s + (3 + 3.31i)20-s + (2.84 − 2.84i)23-s + (0.5 − 4.97i)25-s + (7.15 + 7.15i)27-s − 9.94·31-s + (−7.15 + 7.15i)33-s + 12.6·36-s + (1.47 + 1.47i)37-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s + i·4-s + (0.741 − 0.670i)5-s − 2.10i·9-s + 1.00·11-s + (−1.24 − 1.24i)12-s + (−0.0882 + 1.76i)15-s − 16-s + (0.670 + 0.741i)20-s + (0.592 − 0.592i)23-s + (0.100 − 0.994i)25-s + (1.37 + 1.37i)27-s − 1.78·31-s + (−1.24 + 1.24i)33-s + 2.10·36-s + (0.242 + 0.242i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.547687 + 0.411573i\)
\(L(\frac12)\) \(\approx\) \(0.547687 + 0.411573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.65 + 1.5i)T \)
11 \( 1 - 3.31T \)
good2 \( 1 - 2iT^{2} \)
3 \( 1 + (2.15 - 2.15i)T - 3iT^{2} \)
7 \( 1 - 7iT^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-2.84 + 2.84i)T - 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 9.94T + 31T^{2} \)
37 \( 1 + (-1.47 - 1.47i)T + 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (9.31 + 9.31i)T + 47iT^{2} \)
53 \( 1 + (-3.63 + 3.63i)T - 53iT^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-11.4 - 11.4i)T + 67iT^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 9iT - 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−16.06944287166255069982915769266, −14.63565379371261967531007266738, −13.00672764986797603071920136975, −12.02674481896044705874260636110, −11.06498465641040350573303438256, −9.704271736043385903489154722322, −8.796465424989386929398989851816, −6.60354026877879547142645090208, −5.18591295389126206683905173421, −3.94953060036178666445540372141, 1.63620250690544105536133033406, 5.38481502662120442712443357971, 6.31419279479992790599764988959, 7.15190906861610606750797165916, 9.444252028395126395372971013627, 10.80504971422507305312653999049, 11.48835924044194409813873677265, 12.88317184914451630183209063433, 13.89276280389374815506058098110, 14.84950796954489427052169666659

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