Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} \cdot 7 $
Sign $-0.173 - 0.984i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)7-s + (3 + 5.19i)11-s + (−1 + 1.73i)13-s − 4·19-s + (3 − 5.19i)23-s + (2.5 + 4.33i)25-s + (−3 − 5.19i)29-s + (−4 + 6.92i)31-s + 2·37-s + (−6 + 10.3i)41-s + (2 + 3.46i)43-s + (−6 − 10.3i)47-s + (−0.499 + 0.866i)49-s − 6·53-s + (5 + 8.66i)61-s + ⋯
L(s)  = 1  + (−0.188 − 0.327i)7-s + (0.904 + 1.56i)11-s + (−0.277 + 0.480i)13-s − 0.917·19-s + (0.625 − 1.08i)23-s + (0.5 + 0.866i)25-s + (−0.557 − 0.964i)29-s + (−0.718 + 1.24i)31-s + 0.328·37-s + (−0.937 + 1.62i)41-s + (0.304 + 0.528i)43-s + (−0.875 − 1.51i)47-s + (−0.0714 + 0.123i)49-s − 0.824·53-s + (0.640 + 1.10i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
\( \varepsilon \)  =  $-0.173 - 0.984i$
motivic weight  =  \(1\)
character  :  $\chi_{2268} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2268,\ (\ :1/2),\ -0.173 - 0.984i)\)
\(L(1)\)  \(\approx\)  \(1.264580172\)
\(L(\frac12)\)  \(\approx\)  \(1.264580172\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (6 - 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (-5 - 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.277514125400977374924380205898, −8.581633392470210362959985492589, −7.55487425682306110698043040046, −6.80280541089586687415480976047, −6.45421799154760728205789787926, −5.04670819917669227594888150587, −4.46884447186603265294106039120, −3.61902159725980669399051113580, −2.36346957374264415656614906676, −1.39522684320367510089473238635, 0.43932082541865337390556094970, 1.83537165811994897304706491537, 3.11662560351249626213377973979, 3.71241073276804783745447111211, 4.87651503975233622159056832727, 5.80930879213329643422588999269, 6.31189553496052650422038856896, 7.26718065801961995198100705756, 8.139114171419124658855805662152, 8.916673708942122618435374206025

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