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Properties

Label 2-55-55.43-c1-0-0
Degree $2$
Conductor $55$
Sign $0.983 - 0.180i$
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  + (1.15 + 1.15i)3-s − 2i·4-s + (−1.65 + 1.5i)5-s − 0.316i·9-s − 3.31·11-s + (2.31 − 2.31i)12-s + (−3.65 − 0.183i)15-s − 4·16-s + (3 + 3.31i)20-s + (6.15 + 6.15i)23-s + (0.5 − 4.97i)25-s + (3.84 − 3.84i)27-s + 9.94·31-s + (−3.84 − 3.84i)33-s − 0.633·36-s + (−8.47 + 8.47i)37-s + ⋯
L(s)  = 1  + (0.668 + 0.668i)3-s − i·4-s + (−0.741 + 0.670i)5-s − 0.105i·9-s − 1.00·11-s + (0.668 − 0.668i)12-s + (−0.944 − 0.0473i)15-s − 16-s + (0.670 + 0.741i)20-s + (1.28 + 1.28i)23-s + (0.100 − 0.994i)25-s + (0.739 − 0.739i)27-s + 1.78·31-s + (−0.668 − 0.668i)33-s − 0.105·36-s + (−1.39 + 1.39i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.983 - 0.180i$
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: \((2,\ 55,\ (\ :1/2),\ 0.983 - 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904114 + 0.0823837i\)
\(L(\frac12)\) \(\approx\) \(0.904114 + 0.0823837i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 - 1.15i)T + 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-6.15 - 6.15i)T + 23iT^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + (8.47 - 8.47i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (2.68 - 2.68i)T - 47iT^{2} \)
53 \( 1 + (9.63 + 9.63i)T + 53iT^{2} \)
59 \( 1 + 3.31iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-1.52 + 1.52i)T - 67iT^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (13.4 - 13.4i)T - 97iT^{2} \)
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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

−15.41924105522343092015745758834, −14.54757051505721561843252976477, −13.47315814693768272499698520822, −11.66991069215909497491782236321, −10.53863727231625196018524193354, −9.682303072155755329859385347862, −8.278663877808189586967777173745, −6.70998683208428034530949216352, −4.90586862298137220451443600131, −3.14810190884510550788584453401, 2.85991361577896401400185087078, 4.69283492377536908539427929709, 7.17128803580651590665461554075, 8.083045099302244928948101749412, 8.813174501455526830475471380197, 10.88138989192355342832196302463, 12.32524711808401709605989414965, 12.90018437961591414739990847332, 13.89001808452755247293335082366, 15.46383208499094676616911500459

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