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Properties

Label 2-672-56.19-c1-0-4
Degree $2$
Conductor $672$
Sign $0.605 - 0.795i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.605 - 0.795i$
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: \((2,\ 672,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22360 + 0.606217i\)
\(L(\frac12)\) \(\approx\) \(1.22360 + 0.606217i\)
\(L(\frac{3}{2})\) 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

−10.61856953820574408779863205393, −9.862104213719391272479209719445, −9.031130487893667142331307272785, −7.85515921029056993643257672481, −7.26445799133834276451977619869, −5.98738212636646989822537470633, −5.13018665924932277058396992367, −4.34828938785927618932796653975, −2.95806890788617030947414816895, −1.39383947334189203545483954958, 0.907094920304911866040300292157, 2.37009246956691024507638018673, 3.95347129730172952415015733812, 5.01241437720657565020672551534, 5.72676390975148572658925570809, 6.97643161309481791875852331554, 7.67805931471039292685426749276, 8.516184977040329841180339205966, 9.603269896124714773197746186272, 10.49203892256357033129270325812

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