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Properties

Label 2-2112-88.43-c1-0-23
Degree $2$
Conductor $2112$
Sign $0.144 - 0.989i$
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 + 3.78i·5-s + 3.68·7-s + 9-s + (0.386 + 3.29i)11-s + 5.10·13-s + 3.78i·15-s − 1.27i·17-s + 6.09i·19-s + 3.68·21-s + 2.32i·23-s − 9.34·25-s + 27-s − 3.97·29-s − 8.34i·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.69i·5-s + 1.39·7-s + 0.333·9-s + (0.116 + 0.993i)11-s + 1.41·13-s + 0.978i·15-s − 0.308i·17-s + 1.39i·19-s + 0.804·21-s + 0.484i·23-s − 1.86·25-s + 0.192·27-s − 0.737·29-s − 1.49i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2112\)    =    \(2^{6} \cdot 3 \cdot 11\)
Sign: $0.144 - 0.989i$
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: \((2,\ 2112,\ (\ :1/2),\ 0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.772138714\)
\(L(\frac12)\) \(\approx\) \(2.772138714\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.386 - 3.29i)T \)
good5 \( 1 - 3.78iT - 5T^{2} \)
7 \( 1 - 3.68T + 7T^{2} \)
13 \( 1 - 5.10T + 13T^{2} \)
17 \( 1 + 1.27iT - 17T^{2} \)
19 \( 1 - 6.09iT - 19T^{2} \)
23 \( 1 - 2.32iT - 23T^{2} \)
29 \( 1 + 3.97T + 29T^{2} \)
31 \( 1 + 8.34iT - 31T^{2} \)
37 \( 1 + 6.23iT - 37T^{2} \)
41 \( 1 + 4.90iT - 41T^{2} \)
43 \( 1 + 8.64iT - 43T^{2} \)
47 \( 1 - 3.01iT - 47T^{2} \)
53 \( 1 + 9.33iT - 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 - 15.4T + 61T^{2} \)
67 \( 1 + 0.803T + 67T^{2} \)
71 \( 1 + 8.56iT - 71T^{2} \)
73 \( 1 - 5.39iT - 73T^{2} \)
79 \( 1 + 9.49T + 79T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 - 2.02T + 97T^{2} \)
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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

−9.314280764724000854127181132819, −8.297804086887016989129197203645, −7.66338671488653389145902190878, −7.18021332412566904095450292664, −6.20209510226726919754188562810, −5.39662142638799947727614382221, −4.00894014259505828660595302711, −3.64543553216143617603934472651, −2.29239608194530474127470690698, −1.70541185075550218790700153841, 1.03133206005767645287244964371, 1.57445260653300282773650788268, 3.07766748345200584021743506265, 4.22009297936935293834770100586, 4.78805014112833687980046586764, 5.54585843885543327790926205737, 6.51295667758448249736753664839, 7.79777740996722338383375656015, 8.379585804798683798958797733567, 8.746449594396521908529939316937

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