Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
Sign $0.762 + 0.647i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.46 − 0.916i)3-s + 1.83·5-s i·7-s + (1.31 − 2.69i)9-s + 5.57i·13-s + (2.69 − 1.68i)15-s − 6.08i·17-s + 6.93·19-s + (−0.916 − 1.46i)21-s + 0.697·23-s − 1.63·25-s + (−0.530 − 5.16i)27-s − 7.11·29-s − 1.83i·35-s + 1.87i·37-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)3-s + 0.819·5-s − 0.377i·7-s + (0.439 − 0.898i)9-s + 1.54i·13-s + (0.695 − 0.433i)15-s − 1.47i·17-s + 1.59·19-s + (−0.200 − 0.320i)21-s + 0.145·23-s − 0.327·25-s + (−0.102 − 0.994i)27-s − 1.32·29-s − 0.309i·35-s + 0.308i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 + 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.762 + 0.647i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 672,\ (\ :1/2),\ 0.762 + 0.647i)$
$L(1)$  $\approx$  $2.16420 - 0.794684i$
$L(\frac12)$  $\approx$  $2.16420 - 0.794684i$
$L(\frac{3}{2})$   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 \( 1 + (-1.46 + 0.916i)T \)
7 \( 1 + iT \)
good5 \( 1 - 1.83T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.57iT - 13T^{2} \)
17 \( 1 + 6.08iT - 17T^{2} \)
19 \( 1 - 6.93T + 19T^{2} \)
23 \( 1 - 0.697T + 23T^{2} \)
29 \( 1 + 7.11T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 - 4.69iT - 41T^{2} \)
43 \( 1 + 1.36T + 43T^{2} \)
47 \( 1 + 3.44T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 - 8.94iT - 59T^{2} \)
61 \( 1 + 4.30iT - 61T^{2} \)
67 \( 1 + 4.51T + 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 - 0.438iT - 83T^{2} \)
89 \( 1 - 2.64iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−10.04818017648742293431346462177, −9.418545480007626557162970778803, −8.943280269034082719903967678673, −7.54869855592977136899666285318, −7.11306056421197570292106848749, −6.08215747127900844128130755984, −4.87866160431815062887566365770, −3.63560549803186206627980911089, −2.47493295539165644922585310873, −1.36287198267682676292546522789, 1.74119985657418147964017472877, 2.93727101267638402690796849013, 3.85155765413318304633554302164, 5.35518814365280949593619584118, 5.77465984234024589953729503345, 7.32437039824607383115592522612, 8.119512033134500374849484768754, 8.935950695178778810429955207399, 9.817509569507536125426689648684, 10.28848281916264552717760484104

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