Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
Sign $0.681 - 0.731i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 + 0.448i)3-s + 0.896·5-s + i·7-s + (2.59 − 1.50i)9-s − 1.84i·13-s + (−1.50 + 0.402i)15-s + 4.12i·17-s + 0.654·19-s + (−0.448 − 1.67i)21-s + 7.12·23-s − 4.19·25-s + (−3.67 + 3.67i)27-s + 7.79·29-s + 0.896i·35-s + 10.6i·37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + 0.401·5-s + 0.377i·7-s + (0.865 − 0.500i)9-s − 0.512i·13-s + (−0.387 + 0.103i)15-s + 1.00i·17-s + 0.150·19-s + (−0.0978 − 0.365i)21-s + 1.48·23-s − 0.839·25-s + (−0.706 + 0.707i)27-s + 1.44·29-s + 0.151i·35-s + 1.75i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.681 - 0.731i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 672,\ (\ :1/2),\ 0.681 - 0.731i)$
$L(1)$  $\approx$  $1.06524 + 0.463715i$
$L(\frac12)$  $\approx$  $1.06524 + 0.463715i$
$L(\frac{3}{2})$   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 \( 1 + (1.67 - 0.448i)T \)
7 \( 1 - iT \)
good5 \( 1 - 0.896T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 1.84iT - 13T^{2} \)
17 \( 1 - 4.12iT - 17T^{2} \)
19 \( 1 - 0.654T + 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 - 7.79T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 10.6iT - 37T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 - 1.19T + 43T^{2} \)
47 \( 1 - 9.59T + 47T^{2} \)
53 \( 1 - 8.24T + 53T^{2} \)
59 \( 1 - 6.89iT - 59T^{2} \)
61 \( 1 + 4.54iT - 61T^{2} \)
67 \( 1 - 5.49T + 67T^{2} \)
71 \( 1 + 4.12T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 + 8.99T + 97T^{2} \)
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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

−10.51497134778931937830876379302, −10.00418767757280710199840317327, −9.014705910419124456379531354547, −8.031357383169704931816962497345, −6.82630180901709123196911390379, −6.06513175530223263344677867586, −5.27106114842114213317219144620, −4.32584659545503499749520719630, −2.93999047983456643756400939247, −1.24635816757981625207078040602, 0.844732997191226476839925604834, 2.36510530856770235044761467144, 4.02408298791847760887464944465, 5.05926283687006570173788630001, 5.83867403176755302798954268403, 6.95616673740756664060509744873, 7.36683798951083785508846641470, 8.788708744529045776646838462787, 9.659280278943825041539872853416, 10.52504667229631708530286399748

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