Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
Sign $0.359 - 0.933i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.203 − 1.72i)3-s − 3.44·5-s + i·7-s + (−2.91 − 0.699i)9-s + 5.42i·13-s + (−0.699 + 5.91i)15-s + 3.15i·17-s + 4.40·19-s + (1.72 + 0.203i)21-s + 4.55·23-s + 6.83·25-s + (−1.79 + 4.87i)27-s − 4.08·29-s − 3.44i·35-s + 3.18i·37-s + ⋯
L(s)  = 1  + (0.117 − 0.993i)3-s − 1.53·5-s + 0.377i·7-s + (−0.972 − 0.233i)9-s + 1.50i·13-s + (−0.180 + 1.52i)15-s + 0.765i·17-s + 1.01·19-s + (0.375 + 0.0443i)21-s + 0.949·23-s + 1.36·25-s + (−0.345 + 0.938i)27-s − 0.757·29-s − 0.581i·35-s + 0.523i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.359 - 0.933i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 672,\ (\ :1/2),\ 0.359 - 0.933i)$
$L(1)$  $\approx$  $0.574156 + 0.394139i$
$L(\frac12)$  $\approx$  $0.574156 + 0.394139i$
$L(\frac{3}{2})$   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 \( 1 + (-0.203 + 1.72i)T \)
7 \( 1 - iT \)
good5 \( 1 + 3.44T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.42iT - 13T^{2} \)
17 \( 1 - 3.15iT - 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
23 \( 1 - 4.55T + 23T^{2} \)
29 \( 1 + 4.08T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3.18iT - 37T^{2} \)
41 \( 1 - 5.95iT - 41T^{2} \)
43 \( 1 + 9.83T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 - 6.31T + 53T^{2} \)
59 \( 1 + 0.641iT - 59T^{2} \)
61 \( 1 - 10.2iT - 61T^{2} \)
67 \( 1 - 9.02T + 67T^{2} \)
71 \( 1 + 3.15T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 1.62iT - 79T^{2} \)
83 \( 1 - 12.5iT - 83T^{2} \)
89 \( 1 - 7.80iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−11.17881646353564070564290783587, −9.611671093197260158214742126855, −8.641583620941204715528227287607, −8.065275660770916832811092922513, −7.14672712462825707311455164399, −6.57295705518492788237673829584, −5.22245274060236811464174340290, −4.01599585962053166758974292494, −3.01338555430615761330042736674, −1.47514841662586408117399565735, 0.38624735720477021309646381439, 3.10660813893704458908132572818, 3.59743103504921757231737517813, 4.76268426099529278303626321101, 5.46097789414369192434868379689, 7.08769141266271804656948496810, 7.81855628441496067673722130744, 8.530725098696728093641752996137, 9.553239942595632311120630635560, 10.40106988725070398642618259445

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