Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.28·5-s − 1.96·7-s + 2.40·11-s − 3.92·13-s + 3.63·17-s + 2.30·19-s + 4.55·23-s − 3.34·25-s + 5.81·29-s − 4.65·31-s + 2.52·35-s + 4.09·37-s − 7.34·41-s − 3.88·43-s − 7.51·47-s − 3.14·49-s + 10.6·53-s − 3.09·55-s − 10.8·59-s − 14.0·61-s + 5.04·65-s + 0.286·67-s + 10.1·71-s + 3.44·73-s − 4.73·77-s + 5.99·79-s − 10.0·83-s + ⋯
L(s)  = 1  − 0.574·5-s − 0.742·7-s + 0.726·11-s − 1.08·13-s + 0.882·17-s + 0.529·19-s + 0.948·23-s − 0.669·25-s + 1.07·29-s − 0.836·31-s + 0.426·35-s + 0.672·37-s − 1.14·41-s − 0.592·43-s − 1.09·47-s − 0.448·49-s + 1.46·53-s − 0.417·55-s − 1.40·59-s − 1.79·61-s + 0.625·65-s + 0.0350·67-s + 1.20·71-s + 0.403·73-s − 0.539·77-s + 0.674·79-s − 1.10·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6012} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6012,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.343644188\)
\(L(\frac12)\)  \(\approx\)  \(1.343644188\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 1.28T + 5T^{2} \)
7 \( 1 + 1.96T + 7T^{2} \)
11 \( 1 - 2.40T + 11T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
17 \( 1 - 3.63T + 17T^{2} \)
19 \( 1 - 2.30T + 19T^{2} \)
23 \( 1 - 4.55T + 23T^{2} \)
29 \( 1 - 5.81T + 29T^{2} \)
31 \( 1 + 4.65T + 31T^{2} \)
37 \( 1 - 4.09T + 37T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 + 3.88T + 43T^{2} \)
47 \( 1 + 7.51T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 0.286T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 3.44T + 73T^{2} \)
79 \( 1 - 5.99T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 9.05T + 89T^{2} \)
97 \( 1 - 4.02T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.965026030272994598981312998934, −7.35928790568972555964528322418, −6.75671173174790749250653123079, −6.02940751747733801513910619524, −5.11734530071596428677925369015, −4.48508890222424990811994495222, −3.42126363551032204969124764996, −3.09376458736996657602643132728, −1.81165033501310444938528462250, −0.60523250832563126221853186895, 0.60523250832563126221853186895, 1.81165033501310444938528462250, 3.09376458736996657602643132728, 3.42126363551032204969124764996, 4.48508890222424990811994495222, 5.11734530071596428677925369015, 6.02940751747733801513910619524, 6.75671173174790749250653123079, 7.35928790568972555964528322418, 7.965026030272994598981312998934

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