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Properties

Label 2-2112-264.107-c0-0-3
Degree $2$
Conductor $2112$
Sign $0.514 + 0.857i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 − 0.309i)3-s + (−0.5 − 0.363i)5-s + (0.363 − 1.11i)7-s + (0.809 − 0.587i)9-s + i·11-s + (−0.587 − 0.190i)15-s − 1.17i·21-s + (−0.190 − 0.587i)25-s + (0.587 − 0.809i)27-s + (1.80 + 0.587i)29-s + (−0.951 − 1.30i)31-s + (0.309 + 0.951i)33-s + (−0.587 + 0.427i)35-s − 0.618·45-s + (−0.309 − 0.224i)49-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + (−0.5 − 0.363i)5-s + (0.363 − 1.11i)7-s + (0.809 − 0.587i)9-s + i·11-s + (−0.587 − 0.190i)15-s − 1.17i·21-s + (−0.190 − 0.587i)25-s + (0.587 − 0.809i)27-s + (1.80 + 0.587i)29-s + (−0.951 − 1.30i)31-s + (0.309 + 0.951i)33-s + (−0.587 + 0.427i)35-s − 0.618·45-s + (−0.309 − 0.224i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.514 + 0.857i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2112} (1823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2112,\ (\ :0),\ 0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564479623\)
\(L(\frac12)\) \(\approx\) \(1.564479623\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 - iT \)
good5 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−9.081689988676020273496988483404, −8.226703038523001930238088335494, −7.65888763034506914147899784158, −7.12973699980317318085003185279, −6.25434948487038146258871871822, −4.70200690636752152196733311388, −4.32816188100797040106063487778, −3.40420603163849600625579498385, −2.23088950198636496737200914023, −1.11262528301016465389238580558, 1.71552255934117505154980302933, 2.88700105278525450617234556232, 3.36858668400914192323955545568, 4.50367903586667347583515857028, 5.34327331848315930184995482428, 6.29978909151864485579838537273, 7.26768829954799314749321025735, 8.113415007101558197030885460561, 8.596025732207336072288727791271, 9.180349193516346774415917529269

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