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Properties

Label 2-1122-17.13-c1-0-20
Degree $2$
Conductor $1122$
Sign $0.232 + 0.972i$
Analytic cond. $8.95921$
Root an. cond. $2.99319$
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  − i·2-s + (0.707 + 0.707i)3-s − 4-s + (−0.655 − 0.655i)5-s + (0.707 − 0.707i)6-s + (1.18 − 1.18i)7-s + i·8-s + 1.00i·9-s + (−0.655 + 0.655i)10-s + (0.707 − 0.707i)11-s + (−0.707 − 0.707i)12-s − 0.330·13-s + (−1.18 − 1.18i)14-s − 0.926i·15-s + 16-s + (3.22 + 2.57i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.408 + 0.408i)3-s − 0.5·4-s + (−0.293 − 0.293i)5-s + (0.288 − 0.288i)6-s + (0.446 − 0.446i)7-s + 0.353i·8-s + 0.333i·9-s + (−0.207 + 0.207i)10-s + (0.213 − 0.213i)11-s + (−0.204 − 0.204i)12-s − 0.0916·13-s + (−0.315 − 0.315i)14-s − 0.239i·15-s + 0.250·16-s + (0.781 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $0.232 + 0.972i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ 0.232 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.760947190\)
\(L(\frac12)\) \(\approx\) \(1.760947190\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-3.22 - 2.57i)T \)
good5 \( 1 + (0.655 + 0.655i)T + 5iT^{2} \)
7 \( 1 + (-1.18 + 1.18i)T - 7iT^{2} \)
13 \( 1 + 0.330T + 13T^{2} \)
19 \( 1 + 0.581iT - 19T^{2} \)
23 \( 1 + (-5.39 + 5.39i)T - 23iT^{2} \)
29 \( 1 + (5.18 + 5.18i)T + 29iT^{2} \)
31 \( 1 + (-3.83 - 3.83i)T + 31iT^{2} \)
37 \( 1 + (1.86 + 1.86i)T + 37iT^{2} \)
41 \( 1 + (-7.39 + 7.39i)T - 41iT^{2} \)
43 \( 1 - 6.28iT - 43T^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 + 8.14iT - 53T^{2} \)
59 \( 1 + 2.79iT - 59T^{2} \)
61 \( 1 + (-4.61 + 4.61i)T - 61iT^{2} \)
67 \( 1 + 8.36T + 67T^{2} \)
71 \( 1 + (-3.36 - 3.36i)T + 71iT^{2} \)
73 \( 1 + (-3.87 - 3.87i)T + 73iT^{2} \)
79 \( 1 + (-9.94 + 9.94i)T - 79iT^{2} \)
83 \( 1 + 14.6iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \)
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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

−9.731987753317778689787825689231, −8.875852585636750687806922826933, −8.225233149542781877080751195853, −7.45205779990812423848645260948, −6.18099305499960063589621156609, −4.99513022328246379926479803485, −4.27652681022094971713213081397, −3.44053005133465104822173510863, −2.28887746811810429005479890779, −0.866398278481351230816253768845, 1.33190135005000276970599063824, 2.83634476012845075420230542489, 3.82145211326748136509924635273, 5.07596603062853891666723723550, 5.75514417717273132273854586248, 6.96940295184373933289260239049, 7.44082786484927048489079604133, 8.174528814474501105663843344639, 9.147985612593833127362612060223, 9.600967292015548263547412690275

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