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Properties

Label 2-672-56.27-c1-0-9
Degree $2$
Conductor $672$
Sign $0.999 + 0.0348i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 2.33·5-s + (−0.490 − 2.59i)7-s − 9-s + 0.304·11-s + 5.46·13-s + 2.33i·15-s − 6.37i·17-s + 0.840i·19-s + (2.59 − 0.490i)21-s + 0.111i·23-s + 0.449·25-s − i·27-s + 5.46i·29-s + 7.64·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.04·5-s + (−0.185 − 0.982i)7-s − 0.333·9-s + 0.0918·11-s + 1.51·13-s + 0.602i·15-s − 1.54i·17-s + 0.192i·19-s + (0.567 − 0.107i)21-s + 0.0232i·23-s + 0.0899·25-s − 0.192i·27-s + 1.01i·29-s + 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.999 + 0.0348i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.999 + 0.0348i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83459 - 0.0319972i\)
\(L(\frac12)\) \(\approx\) \(1.83459 - 0.0319972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
7 \( 1 + (0.490 + 2.59i)T \)
good5 \( 1 - 2.33T + 5T^{2} \)
11 \( 1 - 0.304T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 6.37iT - 17T^{2} \)
19 \( 1 - 0.840iT - 19T^{2} \)
23 \( 1 - 0.111iT - 23T^{2} \)
29 \( 1 - 5.46iT - 29T^{2} \)
31 \( 1 - 7.64T + 31T^{2} \)
37 \( 1 + 6.44iT - 37T^{2} \)
41 \( 1 - 8.66iT - 41T^{2} \)
43 \( 1 + 6.06T + 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 + 2.75T + 61T^{2} \)
67 \( 1 - 8.35T + 67T^{2} \)
71 \( 1 + 2.07iT - 71T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 + 7.90iT - 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 - 4.08iT - 89T^{2} \)
97 \( 1 - 6.42iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.49287215270953524542371223508, −9.634344078861507833189600875283, −9.063763346132989583386643896966, −7.943893784496782530944195470385, −6.80054990547009088087690642244, −6.02406391584067537916803522885, −5.00951759751854901319842290304, −3.95338650303010850867765871553, −2.86805474879795915774598063803, −1.19152524252596772948892551575, 1.48925555154643569052573034418, 2.48180667130562381996805746460, 3.82945461776457659323953324699, 5.41964050322216992039181614502, 6.13047444110521408068498890065, 6.59276555970739216161149890867, 8.236748068714121655589524862258, 8.586889421875112020161017987095, 9.642818813085416848510496764973, 10.43586595338415968926012803422

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