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Properties

Label 2-672-12.11-c3-0-18
Degree $2$
Conductor $672$
Sign $-0.728 - 0.685i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.158 + 5.19i)3-s − 4.20i·5-s + 7i·7-s + (−26.9 + 1.64i)9-s + 48.4·11-s − 4.53·13-s + (21.8 − 0.665i)15-s + 81.1i·17-s − 38.4i·19-s + (−36.3 + 1.10i)21-s − 28.7·23-s + 107.·25-s + (−12.8 − 139. i)27-s + 153. i·29-s + 28.0i·31-s + ⋯
L(s)  = 1  + (0.0304 + 0.999i)3-s − 0.375i·5-s + 0.377i·7-s + (−0.998 + 0.0608i)9-s + 1.32·11-s − 0.0968·13-s + (0.375 − 0.0114i)15-s + 1.15i·17-s − 0.464i·19-s + (−0.377 + 0.0115i)21-s − 0.260·23-s + 0.858·25-s + (−0.0912 − 0.995i)27-s + 0.979i·29-s + 0.162i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.728 - 0.685i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ -0.728 - 0.685i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.587212899\)
\(L(\frac12)\) \(\approx\) \(1.587212899\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.158 - 5.19i)T \)
7 \( 1 - 7iT \)
good5 \( 1 + 4.20iT - 125T^{2} \)
11 \( 1 - 48.4T + 1.33e3T^{2} \)
13 \( 1 + 4.53T + 2.19e3T^{2} \)
17 \( 1 - 81.1iT - 4.91e3T^{2} \)
19 \( 1 + 38.4iT - 6.85e3T^{2} \)
23 \( 1 + 28.7T + 1.21e4T^{2} \)
29 \( 1 - 153. iT - 2.43e4T^{2} \)
31 \( 1 - 28.0iT - 2.97e4T^{2} \)
37 \( 1 + 62.7T + 5.06e4T^{2} \)
41 \( 1 + 0.500iT - 6.89e4T^{2} \)
43 \( 1 - 431. iT - 7.95e4T^{2} \)
47 \( 1 + 121.T + 1.03e5T^{2} \)
53 \( 1 - 458. iT - 1.48e5T^{2} \)
59 \( 1 - 307.T + 2.05e5T^{2} \)
61 \( 1 + 727.T + 2.26e5T^{2} \)
67 \( 1 - 173. iT - 3.00e5T^{2} \)
71 \( 1 + 706.T + 3.57e5T^{2} \)
73 \( 1 - 676.T + 3.89e5T^{2} \)
79 \( 1 + 494. iT - 4.93e5T^{2} \)
83 \( 1 + 569.T + 5.71e5T^{2} \)
89 \( 1 - 954. iT - 7.04e5T^{2} \)
97 \( 1 + 624.T + 9.12e5T^{2} \)
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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

−10.44040337284905872849263646893, −9.390944927522381821239343772444, −8.932548854326682093401492960607, −8.140968844020364646142466799110, −6.71692501625785770045027448993, −5.86261124933243542389864186108, −4.81806849885369500357110346826, −4.01762898705405485433068444575, −2.96928021063360962873457197360, −1.38282674698262109972373266618, 0.46513086512710162245172487869, 1.66846558590500564945554876347, 2.90456718683694189474838010763, 4.03284319205533633266241980496, 5.39593171838102407303461941049, 6.51227953891985530839743951066, 6.99231860678452772811505390988, 7.889057880723077845402073956512, 8.858571190384908821849921589529, 9.687139801117583824761262447348

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