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Properties

Label 2-55-55.9-c1-0-2
Degree $2$
Conductor $55$
Sign $0.999 + 0.00123i$
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  + (1.57 + 0.511i)2-s + (−1.16 − 1.59i)3-s + (0.596 + 0.433i)4-s + (0.264 + 2.22i)5-s + (−1.00 − 3.10i)6-s + (−1.31 + 1.81i)7-s + (−1.22 − 1.68i)8-s + (−0.278 + 0.855i)9-s + (−0.718 + 3.62i)10-s + (−3.27 − 0.547i)11-s − 1.45i·12-s + (3.51 + 1.14i)13-s + (−3.00 + 2.18i)14-s + (3.23 − 3.00i)15-s + (−1.52 − 4.68i)16-s + (2.11 − 0.687i)17-s + ⋯
L(s)  = 1  + (1.11 + 0.361i)2-s + (−0.670 − 0.922i)3-s + (0.298 + 0.216i)4-s + (0.118 + 0.992i)5-s + (−0.412 − 1.26i)6-s + (−0.498 + 0.685i)7-s + (−0.434 − 0.597i)8-s + (−0.0926 + 0.285i)9-s + (−0.227 + 1.14i)10-s + (−0.986 − 0.165i)11-s − 0.420i·12-s + (0.975 + 0.317i)13-s + (−0.802 + 0.583i)14-s + (0.836 − 0.774i)15-s + (−0.380 − 1.17i)16-s + (0.513 − 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.999 + 0.00123i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.999 + 0.00123i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09339 - 0.000677137i\)
\(L(\frac12)\) \(\approx\) \(1.09339 - 0.000677137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.264 - 2.22i)T \)
11 \( 1 + (3.27 + 0.547i)T \)
good2 \( 1 + (-1.57 - 0.511i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.16 + 1.59i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (1.31 - 1.81i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.51 - 1.14i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.11 + 0.687i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-4.27 + 3.10i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 3.85iT - 23T^{2} \)
29 \( 1 + (-0.152 - 0.110i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.212 + 0.653i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.52 + 2.09i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.40 - 4.65i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.41iT - 43T^{2} \)
47 \( 1 + (-7.06 - 9.71i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (12.0 + 3.91i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.278 + 0.202i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.535 - 1.64i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 0.650iT - 67T^{2} \)
71 \( 1 + (-1.43 - 4.42i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-5.20 + 7.16i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.23 + 6.88i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.02 + 0.983i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (-2.15 - 0.700i)T + (78.4 + 57.0i)T^{2} \)
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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

−15.32008923409178812925572358357, −13.92799994861780058616453410883, −13.25107423750845982802898807539, −12.20886215024527423332561533025, −11.19050840181903041865234672836, −9.541554176115682212987125679615, −7.38864042952011186591728097562, −6.29734508580839740037113423014, −5.52471605464874888045296550545, −3.20307241903005468498017109637, 3.68106913793495170059861751868, 4.88531348860611381456826658394, 5.78206150697607179360324335768, 8.195376687834184516665686941406, 9.850740454664969184965684809146, 10.84115362219204559961918838295, 12.13735550506668517846453127554, 13.09577041004822237346897685377, 13.86716235296565314120130884013, 15.46865698126782644126371936698

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