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Properties

Label 2-672-12.11-c3-0-15
Degree $2$
Conductor $672$
Sign $-0.00544 - 0.999i$
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  + (−3.69 − 3.65i)3-s + 14.9i·5-s − 7i·7-s + (0.293 + 26.9i)9-s + 22.9·11-s + 21.5·13-s + (54.7 − 55.3i)15-s + 13.7i·17-s − 56.8i·19-s + (−25.5 + 25.8i)21-s − 20.1·23-s − 99.3·25-s + (97.5 − 100. i)27-s + 2.91i·29-s − 61.0i·31-s + ⋯
L(s)  = 1  + (−0.710 − 0.703i)3-s + 1.33i·5-s − 0.377i·7-s + (0.0108 + 0.999i)9-s + 0.628·11-s + 0.459·13-s + (0.942 − 0.952i)15-s + 0.196i·17-s − 0.686i·19-s + (−0.265 + 0.268i)21-s − 0.182·23-s − 0.794·25-s + (0.695 − 0.718i)27-s + 0.0186i·29-s − 0.353i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00544 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.00544 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.00544 - 0.999i$
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: \((2,\ 672,\ (\ :3/2),\ -0.00544 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.128496136\)
\(L(\frac12)\) \(\approx\) \(1.128496136\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.69 + 3.65i)T \)
7 \( 1 + 7iT \)
good5 \( 1 - 14.9iT - 125T^{2} \)
11 \( 1 - 22.9T + 1.33e3T^{2} \)
13 \( 1 - 21.5T + 2.19e3T^{2} \)
17 \( 1 - 13.7iT - 4.91e3T^{2} \)
19 \( 1 + 56.8iT - 6.85e3T^{2} \)
23 \( 1 + 20.1T + 1.21e4T^{2} \)
29 \( 1 - 2.91iT - 2.43e4T^{2} \)
31 \( 1 + 61.0iT - 2.97e4T^{2} \)
37 \( 1 + 157.T + 5.06e4T^{2} \)
41 \( 1 - 408. iT - 6.89e4T^{2} \)
43 \( 1 - 523. iT - 7.95e4T^{2} \)
47 \( 1 + 268.T + 1.03e5T^{2} \)
53 \( 1 + 507. iT - 1.48e5T^{2} \)
59 \( 1 - 684.T + 2.05e5T^{2} \)
61 \( 1 - 151.T + 2.26e5T^{2} \)
67 \( 1 - 48.5iT - 3.00e5T^{2} \)
71 \( 1 + 401.T + 3.57e5T^{2} \)
73 \( 1 + 400.T + 3.89e5T^{2} \)
79 \( 1 - 857. iT - 4.93e5T^{2} \)
83 \( 1 - 5.34T + 5.71e5T^{2} \)
89 \( 1 - 1.58e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.61e3T + 9.12e5T^{2} \)
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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

−10.50011403789610145926467831047, −9.719205508867231937490801265121, −8.355898809216692673833121479441, −7.46703416301257385690165608018, −6.60661537138311964258898005107, −6.28213169501009062166583048873, −4.94786959603756155332215152950, −3.66875933508231623482833552489, −2.49539062966434237054826054258, −1.17039701357562626013920002464, 0.39872156131148872350240876638, 1.60840060943289064122464147652, 3.56045159991724404029645150870, 4.42075615963827564839727984132, 5.34702383246156196214639224890, 5.96361282947449637010761100940, 7.14672592399976361913865690643, 8.648079522806395918374708124220, 8.877854972066087922680744593450, 9.885026414227245721840000276126

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