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Properties

Label 2-2112-24.11-c1-0-30
Degree $2$
Conductor $2112$
Sign $0.639 - 0.769i$
Analytic cond. $16.8644$
Root an. cond. $4.10662$
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  + (1.41 − i)3-s + 2·5-s + 4.44i·7-s + (1.00 − 2.82i)9-s + i·11-s + 3.46i·13-s + (2.82 − 2i)15-s + 4.87i·17-s + 0.778·19-s + (4.44 + 6.29i)21-s − 6.29·23-s − 25-s + (−1.41 − 5.00i)27-s + 5.34·29-s + 6.89i·31-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + 0.894·5-s + 1.68i·7-s + (0.333 − 0.942i)9-s + 0.301i·11-s + 0.960i·13-s + (0.730 − 0.516i)15-s + 1.18i·17-s + 0.178·19-s + (0.970 + 1.37i)21-s − 1.31·23-s − 0.200·25-s + (−0.272 − 0.962i)27-s + 0.993·29-s + 1.23i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.639 - 0.769i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 0.639 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.644449765\)
\(L(\frac12)\) \(\approx\) \(2.644449765\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.41 + i)T \)
11 \( 1 - iT \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 - 4.44iT - 7T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 4.87iT - 17T^{2} \)
19 \( 1 - 0.778T + 19T^{2} \)
23 \( 1 + 6.29T + 23T^{2} \)
29 \( 1 - 5.34T + 29T^{2} \)
31 \( 1 - 6.89iT - 31T^{2} \)
37 \( 1 - 2.19iT - 37T^{2} \)
41 \( 1 + 7.70iT - 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2.89T + 53T^{2} \)
59 \( 1 - 9.79iT - 59T^{2} \)
61 \( 1 + 9.12iT - 61T^{2} \)
67 \( 1 + 9.75T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 - 6.89T + 73T^{2} \)
79 \( 1 + 9.34iT - 79T^{2} \)
83 \( 1 - 9.79iT - 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−8.981669806350702781514718854700, −8.648459615070704516356236432196, −7.83317475552918430965849585296, −6.69022056823308288705117156071, −6.16319141848108334524973801434, −5.45507418695723043457559453410, −4.25568907207394550407837375599, −3.11696989155151103213255535696, −2.07000817107493763300059651102, −1.78037916855648159243815129263, 0.813473156736066260456472951044, 2.19585265629606278500265592230, 3.17706765477617536544228281817, 4.02333627599372337938822571447, 4.81399667019817835641853241022, 5.75329801299096214803486478490, 6.73157856921645746275271082455, 7.72791281244050089864153257615, 8.009981916662377241930101886834, 9.189861662460443470790069272147

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