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Properties

Label 2-672-56.19-c1-0-12
Degree $2$
Conductor $672$
Sign $-0.781 + 0.623i$
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 + (1.25 − 2.16i)5-s + (−1.36 + 2.26i)7-s + (0.499 − 0.866i)9-s + (−2.83 − 4.91i)11-s − 5.31·13-s + 2.50i·15-s + (−0.393 + 0.227i)17-s + (3.19 + 1.84i)19-s + (0.0468 − 2.64i)21-s + (−4.43 − 2.56i)23-s + (−0.632 − 1.09i)25-s + 0.999i·27-s + 2.57i·29-s + (−3.00 − 5.20i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (0.559 − 0.969i)5-s + (−0.515 + 0.857i)7-s + (0.166 − 0.288i)9-s + (−0.855 − 1.48i)11-s − 1.47·13-s + 0.646i·15-s + (−0.0955 + 0.0551i)17-s + (0.733 + 0.423i)19-s + (0.0102 − 0.577i)21-s + (−0.924 − 0.533i)23-s + (−0.126 − 0.219i)25-s + 0.192i·27-s + 0.479i·29-s + (−0.539 − 0.934i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.781 + 0.623i$
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: \((2,\ 672,\ (\ :1/2),\ -0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159577 - 0.456121i\)
\(L(\frac12)\) \(\approx\) \(0.159577 - 0.456121i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 - 2.26i)T \)
good5 \( 1 + (-1.25 + 2.16i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.83 + 4.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + (0.393 - 0.227i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.19 - 1.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.43 + 2.56i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.57iT - 29T^{2} \)
31 \( 1 + (3.00 + 5.20i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.80 + 4.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 + 3.66T + 43T^{2} \)
47 \( 1 + (-0.478 + 0.829i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.41 - 3.12i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.76 + 5.06i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.50 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.65 + 8.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.35iT - 71T^{2} \)
73 \( 1 + (-5.93 + 3.42i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.71 - 4.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.96iT - 83T^{2} \)
89 \( 1 + (-5.91 - 3.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.71iT - 97T^{2} \)
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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

−10.06307755200666808106684096465, −9.340962442208373876664833180639, −8.636420908522667698128573926082, −7.63634977349266488624410901532, −6.26741419845245382038829980489, −5.44658790478844967985272695670, −5.06820873437258048288985987778, −3.48037033637103377573428711382, −2.19693282754403382039321667774, −0.25069037315480310520472406251, 1.98269987726626352327466530886, 3.04489359472708680585645188151, 4.56508097149730403493646785807, 5.41170037808820564443258761611, 6.74446200341629925256093099284, 7.09600359403983249592444295700, 7.84185176897033095407021111391, 9.642810329716240073335336265651, 10.03231584374475171913736387167, 10.57008972196137505891647038159

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