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Properties

Label 2-110-55.43-c1-0-2
Degree $2$
Conductor $110$
Sign $0.545 - 0.838i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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  + (0.707 + 0.707i)2-s + (0.292 + 0.292i)3-s + 1.00i·4-s + (0.707 + 2.12i)5-s + 0.414i·6-s + (−2.12 − 2.12i)7-s + (−0.707 + 0.707i)8-s − 2.82i·9-s + (−0.999 + 2i)10-s + (1.41 + 3i)11-s + (−0.292 + 0.292i)12-s + (3 − 3i)13-s − 3i·14-s + (−0.414 + 0.828i)15-s − 1.00·16-s + (−0.878 − 0.878i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.169 + 0.169i)3-s + 0.500i·4-s + (0.316 + 0.948i)5-s + 0.169i·6-s + (−0.801 − 0.801i)7-s + (−0.250 + 0.250i)8-s − 0.942i·9-s + (−0.316 + 0.632i)10-s + (0.426 + 0.904i)11-s + (−0.0845 + 0.0845i)12-s + (0.832 − 0.832i)13-s − 0.801i·14-s + (−0.106 + 0.213i)15-s − 0.250·16-s + (−0.213 − 0.213i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.545 - 0.838i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17909 + 0.639621i\)
\(L(\frac12)\) \(\approx\) \(1.17909 + 0.639621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 2.12i)T \)
11 \( 1 + (-1.41 - 3i)T \)
good3 \( 1 + (-0.292 - 0.292i)T + 3iT^{2} \)
7 \( 1 + (2.12 + 2.12i)T + 7iT^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (0.878 + 0.878i)T + 17iT^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + (5.82 + 5.82i)T + 23iT^{2} \)
29 \( 1 - 7.24T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + (4.12 - 4.12i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (7.24 - 7.24i)T - 43iT^{2} \)
47 \( 1 + (-1.58 + 1.58i)T - 47iT^{2} \)
53 \( 1 + (-2.46 - 2.46i)T + 53iT^{2} \)
59 \( 1 - 1.41iT - 59T^{2} \)
61 \( 1 + 1.24iT - 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 + (-6 + 6i)T - 73iT^{2} \)
79 \( 1 - 1.75T + 79T^{2} \)
83 \( 1 + (-1.24 + 1.24i)T - 83iT^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (-6.24 + 6.24i)T - 97iT^{2} \)
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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

−13.94899837191289466138790556587, −13.01695573167967684617661521812, −11.97009170806135149141438759919, −10.47587401765433130363582167783, −9.765061844068366462901407380717, −8.234419204157333859961660367458, −6.66982817656714865729393670030, −6.38135762585198493384827002430, −4.24691816132634715133386340351, −3.09814769631830291355157683830, 2.00281678291443671113746900445, 3.85074730143178893186960108990, 5.41647617266707158627460822550, 6.37611975826394405004313110237, 8.422900226516719833074242586972, 9.128431095279566032597316304171, 10.42010946134101646885186404052, 11.69325087857158178823418263557, 12.50468959796832512801882494495, 13.61875722828907539456556723254

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