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Properties

Label 2-2112-264.197-c1-0-59
Degree $2$
Conductor $2112$
Sign $0.472 + 0.881i$
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  + (−1.65 − 0.5i)3-s + 3·5-s + (2.5 + 1.65i)9-s − 3.31·11-s + (−4.97 − 1.5i)15-s − 9i·23-s + 4·25-s + (−3.31 − 4i)27-s + 9.94·31-s + (5.5 + 1.65i)33-s − 9.94i·37-s + (7.5 + 4.97i)45-s + 12i·47-s + 7·49-s − 6·53-s + ⋯
L(s)  = 1  + (−0.957 − 0.288i)3-s + 1.34·5-s + (0.833 + 0.552i)9-s − 1.00·11-s + (−1.28 − 0.387i)15-s − 1.87i·23-s + 0.800·25-s + (−0.638 − 0.769i)27-s + 1.78·31-s + (0.957 + 0.288i)33-s − 1.63i·37-s + (1.11 + 0.741i)45-s + 1.75i·47-s + 49-s − 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.472 + 0.881i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.446925062\)
\(L(\frac12)\) \(\approx\) \(1.446925062\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 + 0.5i)T \)
11 \( 1 + 3.31T \)
good5 \( 1 - 3T + 5T^{2} \)
7 \( 1 - 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.94T + 31T^{2} \)
37 \( 1 + 9.94iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 3.31T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 3iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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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

−9.077244321371033493944525838214, −8.114128191259054271289747052394, −7.28585513680693509116975294593, −6.27262877725639534841066011366, −6.01052717103048903051056508740, −5.05349295046920248260008110624, −4.44033260490343137021421261708, −2.73797337309425277775698518710, −1.98989279927646202130256757757, −0.64775156322734916349604737346, 1.13288604239763917716737707956, 2.26074205494870076190728034418, 3.42960181470317596326110441534, 4.71897021818221406725354986223, 5.34368273453212382061853898875, 5.93436466935571743994124194222, 6.66768004072359301402578692841, 7.54827371675536541084674199676, 8.557320057025257593100319907906, 9.560423157383604247572241066210

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