Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
Sign $0.245 + 0.969i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.245 + 0.969i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (239, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 672,\ (\ :1/2),\ 0.245 + 0.969i)$
$L(1)$  $\approx$  $1.32713 - 1.03255i$
$L(\frac12)$  $\approx$  $1.32713 - 1.03255i$
$L(\frac{3}{2})$   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

−10.11602857842560233889012390313, −9.398753442956138463430400684169, −8.399291639477324709908574378153, −7.72541792240700070343965472271, −7.16123461275504788654591494015, −5.85157137256237005584082640631, −4.74113048824370531416053304531, −3.34121715143151803048464949006, −2.75562286562231778517233645325, −0.875898040025905573793033105663, 1.74235581112556368978813290084, 3.29884625288680330731072295097, 4.04674965616463584013441953947, 4.85732857997018873144706272696, 6.38017055963432306673091409112, 7.43496544707878537275151412407, 8.090123795820785490283900203283, 8.951509898954055660444534391197, 9.759636986532578285983250304625, 10.58249383040139515946087512320

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