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  :  $\chi_{2268} (1513, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2268,\ (\ :1/2),\ -0.173 + 0.984i)\)
\(L(1)\)  \(\approx\)  \(0.7966260591\)
\(L(\frac12)\)  \(\approx\)  \(0.7966260591\)
\(L(\frac{3}{2})\)   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

−8.714856379864109155626557695190, −8.025460003388224801738524162685, −7.34087334667278035791086019937, −6.45319460686884828453882081888, −5.69729846878226190519614019299, −4.66826700627741717419464487389, −4.15706293341624450597361385327, −2.59775249824722536351258407399, −2.19555793939704123652728816113, −0.28194619615959937496546034655, 1.20751825795538845446611096919, 2.61454819182830619596687862796, 3.43994453101223211628149269579, 4.35508326462730648850119497692, 5.47005332806805922470972451189, 5.94208133767132300212196196801, 7.06485535008402857043182799627, 7.58700247885894869043234030820, 8.691708748413608990135737158582, 8.954083983417968950778595705529

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