Properties

Degree $2$
Conductor $2672$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.30·3-s − 3·5-s + 0.302·7-s − 1.30·9-s + 2.69·13-s − 3.90·15-s + 2.30·17-s − 2·19-s + 0.394·21-s + 2.30·23-s + 4·25-s − 5.60·27-s + 7.60·29-s − 6.60·31-s − 0.908·35-s + 0.394·37-s + 3.51·39-s − 6.21·41-s − 9.60·43-s + 3.90·45-s − 1.60·47-s − 6.90·49-s + 3·51-s − 4.60·53-s − 2.60·57-s − 7.81·59-s + 6.81·61-s + ⋯
L(s)  = 1  + 0.752·3-s − 1.34·5-s + 0.114·7-s − 0.434·9-s + 0.748·13-s − 1.00·15-s + 0.558·17-s − 0.458·19-s + 0.0860·21-s + 0.480·23-s + 0.800·25-s − 1.07·27-s + 1.41·29-s − 1.18·31-s − 0.153·35-s + 0.0648·37-s + 0.562·39-s − 0.970·41-s − 1.46·43-s + 0.582·45-s − 0.234·47-s − 0.986·49-s + 0.420·51-s − 0.632·53-s − 0.345·57-s − 1.01·59-s + 0.872·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2672\)    =    \(2^{4} \cdot 167\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{2672} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2672,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
167 \( 1 - T \)
good3 \( 1 - 1.30T + 3T^{2} \)
5 \( 1 + 3T + 5T^{2} \)
7 \( 1 - 0.302T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 - 2.30T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 2.30T + 23T^{2} \)
29 \( 1 - 7.60T + 29T^{2} \)
31 \( 1 + 6.60T + 31T^{2} \)
37 \( 1 - 0.394T + 37T^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 + 9.60T + 43T^{2} \)
47 \( 1 + 1.60T + 47T^{2} \)
53 \( 1 + 4.60T + 53T^{2} \)
59 \( 1 + 7.81T + 59T^{2} \)
61 \( 1 - 6.81T + 61T^{2} \)
67 \( 1 + 8.21T + 67T^{2} \)
71 \( 1 + 3.90T + 71T^{2} \)
73 \( 1 + 3.09T + 73T^{2} \)
79 \( 1 + 6.39T + 79T^{2} \)
83 \( 1 - 1.60T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 - 4.51T + 97T^{2} \)
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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

−8.382725397743591804389177484016, −7.958114423324402655419227903907, −7.13568508343499727524962763224, −6.28536557366232249434706804624, −5.22514132203385272999856377627, −4.31283093638949506405694297817, −3.45989368881514867852212270543, −2.98178009582694149725519491649, −1.55807522310230759260450286225, 0, 1.55807522310230759260450286225, 2.98178009582694149725519491649, 3.45989368881514867852212270543, 4.31283093638949506405694297817, 5.22514132203385272999856377627, 6.28536557366232249434706804624, 7.13568508343499727524962763224, 7.958114423324402655419227903907, 8.382725397743591804389177484016

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