Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
Sign $-0.956 + 0.292i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.956 + 0.292i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 672,\ (\ :1/2),\ -0.956 + 0.292i)$
$L(1)$  $\approx$  $0.0326480 - 0.217993i$
$L(\frac12)$  $\approx$  $0.0326480 - 0.217993i$
$L(\frac{3}{2})$   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

−10.14647988250190226042605358728, −9.465547511041898865906455259209, −8.123770581389422679872434149331, −7.50271497150445896976137398062, −6.44836356625613118319979124335, −5.60346466826902836430261578389, −4.75509027620398015954465387762, −3.52996846413259460117574161091, −1.92716004911322060342483326951, −0.12996123021008500749477993652, 1.72102239180814825761060095540, 3.75201102225018579089186272708, 4.31538456145551234006297725094, 5.59852080415097180576293122194, 6.31254793686027730153747369478, 7.35225759834224660768387313078, 8.146906894269722684385562986448, 9.453446709422517233016835702336, 10.06560133324475624358190492278, 11.07315321878206461738434875512

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