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Properties

Label 2-55-11.5-c1-0-3
Degree $2$
Conductor $55$
Sign $-0.817 + 0.576i$
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  + (−0.647 − 1.99i)2-s + (−1.54 − 1.12i)3-s + (−1.93 + 1.40i)4-s + (−0.309 + 0.951i)5-s + (−1.23 + 3.81i)6-s + (2.48 − 1.80i)7-s + (0.661 + 0.480i)8-s + (0.203 + 0.627i)9-s + 2.09·10-s + (1.86 − 2.74i)11-s + 4.57·12-s + (0.942 + 2.90i)13-s + (−5.19 − 3.77i)14-s + (1.54 − 1.12i)15-s + (−0.947 + 2.91i)16-s + (−0.143 + 0.441i)17-s + ⋯
L(s)  = 1  + (−0.457 − 1.40i)2-s + (−0.893 − 0.649i)3-s + (−0.966 + 0.702i)4-s + (−0.138 + 0.425i)5-s + (−0.505 + 1.55i)6-s + (0.937 − 0.681i)7-s + (0.233 + 0.169i)8-s + (0.0679 + 0.209i)9-s + 0.662·10-s + (0.561 − 0.827i)11-s + 1.31·12-s + (0.261 + 0.804i)13-s + (−1.38 − 1.00i)14-s + (0.399 − 0.290i)15-s + (−0.236 + 0.729i)16-s + (−0.0347 + 0.106i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.817 + 0.576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $-0.817 + 0.576i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ -0.817 + 0.576i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.169046 - 0.532968i\)
\(L(\frac12)\) \(\approx\) \(0.169046 - 0.532968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-1.86 + 2.74i)T \)
good2 \( 1 + (0.647 + 1.99i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.54 + 1.12i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-2.48 + 1.80i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.942 - 2.90i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.143 - 0.441i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-6.38 - 4.64i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 + (3.01 - 2.18i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.23 + 9.96i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.49 - 1.08i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.56 - 2.59i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.31T + 43T^{2} \)
47 \( 1 + (-2.41 - 1.75i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.29 - 3.98i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.27 - 1.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.623 - 1.91i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + (2.01 - 6.20i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.98 + 5.80i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.57 + 10.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.75 + 8.48i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 + (-4.74 - 14.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.53416320512772090251843737370, −13.43900384111408200570983337714, −11.96126595051908401810950714718, −11.49058294579171454424681835488, −10.72499956925593599959934285552, −9.286594050091118989655901646978, −7.63011096830609193030911337742, −6.06527698036557441781340316570, −3.78937956558104829296424590744, −1.34817651039807396988066591887, 4.91783809892742614682988910210, 5.58088528739205081050769755339, 7.24657486248023277819964278449, 8.489606551711624541755979738950, 9.611613572256112808500504659536, 11.19544299008499106645909773211, 12.17423191772812422805610430146, 14.04599042829901907666434106891, 15.25214827528612581391590624419, 15.79023999116333882000692788610

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