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Properties

Label 2-672-56.19-c1-0-3
Degree $2$
Conductor $672$
Sign $0.880 - 0.474i$
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  + (−0.866 + 0.5i)3-s + (0.155 − 0.268i)5-s + (−2.58 − 0.560i)7-s + (0.499 − 0.866i)9-s + (0.622 + 1.07i)11-s + 2.68·13-s + 0.310i·15-s + (1.93 − 1.11i)17-s + (5.14 + 2.96i)19-s + (2.51 − 0.807i)21-s + (2.86 + 1.65i)23-s + (2.45 + 4.24i)25-s + 0.999i·27-s + 0.191i·29-s + (−1.95 − 3.38i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.0694 − 0.120i)5-s + (−0.977 − 0.211i)7-s + (0.166 − 0.288i)9-s + (0.187 + 0.325i)11-s + 0.745·13-s + 0.0801i·15-s + (0.468 − 0.270i)17-s + (1.17 + 0.681i)19-s + (0.549 − 0.176i)21-s + (0.596 + 0.344i)23-s + (0.490 + 0.849i)25-s + 0.192i·27-s + 0.0356i·29-s + (−0.351 − 0.608i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.880 - 0.474i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.880 - 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17277 + 0.295941i\)
\(L(\frac12)\) \(\approx\) \(1.17277 + 0.295941i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.58 + 0.560i)T \)
good5 \( 1 + (-0.155 + 0.268i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.622 - 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.68T + 13T^{2} \)
17 \( 1 + (-1.93 + 1.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.14 - 2.96i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.86 - 1.65i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.191iT - 29T^{2} \)
31 \( 1 + (1.95 + 3.38i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.643 - 0.371i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.28iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + (-5.43 + 9.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.8 - 6.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.16 - 2.98i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.58 + 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.25 - 3.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.92iT - 71T^{2} \)
73 \( 1 + (-6.97 + 4.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.2 - 7.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (5.53 + 3.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.62iT - 97T^{2} \)
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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

−10.57470810394636020709192386946, −9.576872651054775666963068751395, −9.253568270973224283856210638580, −7.83378613166782835262718488394, −6.97273363088437782868574238836, −6.02097769876032109207762849265, −5.24520827886411180849690126401, −3.96403119945134201733705850344, −3.09135533588140122629043211155, −1.12972317802018470615619503389, 0.900437516697006342149315536905, 2.71004742749111636203177289391, 3.75346360218589532139230452742, 5.15166512282647325899072301020, 6.06402133281245406811150687213, 6.73657229966675440442683381450, 7.68515088725328484618529388005, 8.860261944913257359827089003659, 9.518733244537155295848498034318, 10.61843610782823364262169321379

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