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Properties

Label 2-55-55.9-c1-0-0
Degree $2$
Conductor $55$
Sign $0.925 - 0.379i$
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 + (1.09 + 1.95i)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 + (−1.85 + 4.00i)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 + (0.487 + 0.872i)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 + (−0.478 + 1.03i)15-s + (−0.380 − 1.17i)16-s + (−0.513 + 0.166i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.925 - 0.379i$
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: \((2,\ 55,\ (\ :1/2),\ 0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.620659 + 0.122428i\)
\(L(\frac12)\) \(\approx\) \(0.620659 + 0.122428i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.09 - 1.95i)T \)
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

−15.31739674479567226383989673301, −14.44765552192385950520942124573, −13.48801398897690124601427867420, −11.30047409475135148728625725799, −10.26936978922791316154156721607, −9.858280977346761509250954419619, −8.517618546926104258793652664419, −7.28813756971085015803887982344, −4.87741691774673586362181483717, −2.75212068296630253530530737040, 1.94878907718246307495164978206, 5.18879693533558404290212401713, 7.24993341575963697579813974643, 8.122166680079673835951933274744, 9.004208728143564498077365866155, 10.08665146896318619022935562280, 12.08219319194279068195359785527, 13.09064029434752798049311931061, 13.97265281900756522143056943120, 15.52524714297895574455526593364

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