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Properties

Label 2-2112-88.43-c1-0-29
Degree $2$
Conductor $2112$
Sign $0.844 + 0.535i$
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  − 3-s + 1.23i·5-s + 3.80·7-s + 9-s + (−3.23 + 0.726i)11-s + 2.35·13-s − 1.23i·15-s − 4.70i·17-s − 6.15i·19-s − 3.80·21-s − 3.23i·23-s + 3.47·25-s − 27-s − 1.45·29-s − 4.47i·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.552i·5-s + 1.43·7-s + 0.333·9-s + (−0.975 + 0.219i)11-s + 0.652·13-s − 0.319i·15-s − 1.14i·17-s − 1.41i·19-s − 0.830·21-s − 0.674i·23-s + 0.694·25-s − 0.192·27-s − 0.269·29-s − 0.803i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(16.8644\)
Root analytic conductor: \(4.10662\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :1/2),\ 0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.594769904\)
\(L(\frac12)\) \(\approx\) \(1.594769904\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 + (3.23 - 0.726i)T \)
good5 \( 1 - 1.23iT - 5T^{2} \)
7 \( 1 - 3.80T + 7T^{2} \)
13 \( 1 - 2.35T + 13T^{2} \)
17 \( 1 + 4.70iT - 17T^{2} \)
19 \( 1 + 6.15iT - 19T^{2} \)
23 \( 1 + 3.23iT - 23T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 + 4.47iT - 31T^{2} \)
37 \( 1 + 2.47iT - 37T^{2} \)
41 \( 1 - 7.60iT - 41T^{2} \)
43 \( 1 - 1.45iT - 43T^{2} \)
47 \( 1 + 0.763iT - 47T^{2} \)
53 \( 1 - 1.23iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 9.95T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 13.7iT - 71T^{2} \)
73 \( 1 - 7.60iT - 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 9.06iT - 83T^{2} \)
89 \( 1 - 9.41T + 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−8.967010484808882620352251392681, −8.158605207537562780431187873210, −7.41719877479195978694776649199, −6.79318231687352017096279334081, −5.76931287980309730459112072912, −4.90110928886563776149062632297, −4.51366444619928753027747579356, −3.01346701929530867579331128017, −2.12397552767089335905003544555, −0.70650284639512759314809994900, 1.16013704122978724273032053679, 1.94879061938830197245101717653, 3.55808326859378872372549383487, 4.42524788697848444731639467702, 5.38441176776223817287482212543, 5.63856312045354208965959059334, 6.81278364768360591924933984125, 7.910107347994371770641980762690, 8.221473117512416319637634324392, 8.966927110435795595136884491429

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