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Properties

Label 2-672-168.11-c1-0-5
Degree $2$
Conductor $672$
Sign $0.678 - 0.734i$
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  + (0.316 − 1.70i)3-s + (1.86 + 3.22i)5-s + (−2.46 + 0.962i)7-s + (−2.79 − 1.07i)9-s + (2.26 + 1.31i)11-s + 3.57i·13-s + (6.07 − 2.14i)15-s + (0.186 + 0.107i)17-s + (1.14 + 1.97i)19-s + (0.859 + 4.50i)21-s + (2.33 + 4.04i)23-s + (−4.42 + 7.66i)25-s + (−2.72 + 4.42i)27-s + 2.57·29-s + (−4.26 − 2.46i)31-s + ⋯
L(s)  = 1  + (0.182 − 0.983i)3-s + (0.832 + 1.44i)5-s + (−0.931 + 0.363i)7-s + (−0.933 − 0.359i)9-s + (0.684 + 0.395i)11-s + 0.990i·13-s + (1.56 − 0.554i)15-s + (0.0451 + 0.0260i)17-s + (0.261 + 0.453i)19-s + (0.187 + 0.982i)21-s + (0.487 + 0.843i)23-s + (−0.885 + 1.53i)25-s + (−0.524 + 0.851i)27-s + 0.478·29-s + (−0.765 − 0.441i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.678 - 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42752 + 0.625211i\)
\(L(\frac12)\) \(\approx\) \(1.42752 + 0.625211i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.316 + 1.70i)T \)
7 \( 1 + (2.46 - 0.962i)T \)
good5 \( 1 + (-1.86 - 3.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.26 - 1.31i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.57iT - 13T^{2} \)
17 \( 1 + (-0.186 - 0.107i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.14 - 1.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.33 - 4.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.57T + 29T^{2} \)
31 \( 1 + (4.26 + 2.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.31 + 4.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.909iT - 41T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + (-0.586 - 1.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.06 - 1.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.79 - 3.92i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.301 - 0.173i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.98 + 8.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + (-4.45 + 7.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.70 + 5.60i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.73iT - 83T^{2} \)
89 \( 1 + (-15.4 + 8.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−10.64053312172973520116860807123, −9.495310808896407264747844078798, −9.237476651327440074771035814908, −7.71275807623404726001257815914, −6.86108483253359178293417861492, −6.42921705045033964863166705662, −5.64867976321762334186168124421, −3.69146772995897822829683820435, −2.72334105312110967237550723720, −1.77577329997217696398045183610, 0.837580036693103826920049868236, 2.76734572074057912662956547442, 3.90008674401551924220999918728, 4.93194226094738519733461499461, 5.64970702779505212035617438907, 6.62109817601159186987307598614, 8.185360816287669136332415492495, 8.868333792975095043122908977952, 9.549011776402230384470640112036, 10.10783388076486413057338976661

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