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Properties

Label 2-1122-33.32-c1-0-47
Degree $2$
Conductor $1122$
Sign $-0.323 + 0.946i$
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  + 2-s + (−0.423 − 1.67i)3-s + 4-s + 4.07i·5-s + (−0.423 − 1.67i)6-s − 4.79i·7-s + 8-s + (−2.64 + 1.42i)9-s + 4.07i·10-s + (−2.78 − 1.80i)11-s + (−0.423 − 1.67i)12-s − 2.57i·13-s − 4.79i·14-s + (6.83 − 1.72i)15-s + 16-s − 17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.244 − 0.969i)3-s + 0.5·4-s + 1.82i·5-s + (−0.172 − 0.685i)6-s − 1.81i·7-s + 0.353·8-s + (−0.880 + 0.474i)9-s + 1.28i·10-s + (−0.838 − 0.544i)11-s + (−0.122 − 0.484i)12-s − 0.713i·13-s − 1.28i·14-s + (1.76 − 0.445i)15-s + 0.250·16-s − 0.242·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1122\)    =    \(2 \cdot 3 \cdot 11 \cdot 17\)
Sign: $-0.323 + 0.946i$
Analytic conductor: \(8.95921\)
Root analytic conductor: \(2.99319\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1122} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1122,\ (\ :1/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.765499983\)
\(L(\frac12)\) \(\approx\) \(1.765499983\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (0.423 + 1.67i)T \)
11 \( 1 + (2.78 + 1.80i)T \)
17 \( 1 + T \)
good5 \( 1 - 4.07iT - 5T^{2} \)
7 \( 1 + 4.79iT - 7T^{2} \)
13 \( 1 + 2.57iT - 13T^{2} \)
19 \( 1 + 4.54iT - 19T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 - 7.91T + 29T^{2} \)
31 \( 1 + 0.355T + 31T^{2} \)
37 \( 1 - 5.35T + 37T^{2} \)
41 \( 1 - 1.52T + 41T^{2} \)
43 \( 1 + 4.45iT - 43T^{2} \)
47 \( 1 + 9.23iT - 47T^{2} \)
53 \( 1 + 9.72iT - 53T^{2} \)
59 \( 1 - 11.6iT - 59T^{2} \)
61 \( 1 - 4.09iT - 61T^{2} \)
67 \( 1 + 5.11T + 67T^{2} \)
71 \( 1 - 7.91iT - 71T^{2} \)
73 \( 1 + 4.26iT - 73T^{2} \)
79 \( 1 - 9.05iT - 79T^{2} \)
83 \( 1 + 7.26T + 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 - 0.0605T + 97T^{2} \)
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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

−10.25209814195537598643546135362, −8.272399059401747428300485827979, −7.57871240949892511411504055402, −6.81800883153895468725115486109, −6.61161758811060008276132269857, −5.46298453011146118094665999859, −4.21350759700272542619228500741, −3.06467757968460324938701309844, −2.51373458850551058404690285933, −0.60882104418972998167187851216, 1.76810191379294891971500099326, 2.98443499636693358285559064883, 4.42009501753338576640177611539, 4.81108559814920718522858571599, 5.65589997927887491115469026942, 6.07229952556695357679070533580, 7.914014868141629881424217616261, 8.551350131533394869145465584953, 9.386841674844843878991138060159, 9.754396894571961099421613004001

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