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Properties

Label 2-2112-88.43-c1-0-35
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

\(L(1)\) \(\approx\) \(2.754299182\)
\(L(\frac12)\) \(\approx\) \(2.754299182\)
\(L(\frac{3}{2})\) 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

−8.936862008163267681199208501172, −8.245274199861054925163200406233, −7.71672335648408650779811583154, −6.82883759927690343050073126341, −5.76636422992353834617463971786, −4.75191786639543247065829209155, −4.36973379613151281840511909872, −3.13397924312298050523042460835, −1.97756425585687873960851387364, −1.06846437398289278528684945330, 1.35303620115156347109209667354, 2.24077778079260068514273026912, 3.30760033528959452075878587275, 4.34659302056030982951272936176, 4.95064090878028718079497118573, 6.09258021015424141942676179565, 7.14202950034836298629517801001, 7.45007340646719702111458055384, 8.611536729740886108103269960704, 8.874442139059368244550794894873

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