Properties

Degree $2$
Conductor $672$
Sign $-0.816 - 0.577i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.292 + 1.70i)3-s + 3.41i·5-s + i·7-s + (−2.82 + i)9-s + 2·11-s + 3.41·13-s + (−5.82 + i)15-s + 4.82i·17-s − 6.24i·19-s + (−1.70 + 0.292i)21-s − 4·23-s − 6.65·25-s + (−2.53 − 4.53i)27-s + 4.82i·29-s + 1.17i·31-s + ⋯
L(s)  = 1  + (0.169 + 0.985i)3-s + 1.52i·5-s + 0.377i·7-s + (−0.942 + 0.333i)9-s + 0.603·11-s + 0.946·13-s + (−1.50 + 0.258i)15-s + 1.17i·17-s − 1.43i·19-s + (−0.372 + 0.0639i)21-s − 0.834·23-s − 1.33·25-s + (−0.487 − 0.872i)27-s + 0.896i·29-s + 0.210i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.816 - 0.577i$
Motivic weight: \(1\)
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.448578 + 1.41134i\)
\(L(\frac12)\) \(\approx\) \(0.448578 + 1.41134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 1.70i)T \)
7 \( 1 - iT \)
good5 \( 1 - 3.41iT - 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 - 4.82iT - 17T^{2} \)
19 \( 1 + 6.24iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4.82iT - 29T^{2} \)
31 \( 1 - 1.17iT - 31T^{2} \)
37 \( 1 + 0.828T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 + 10.4iT - 43T^{2} \)
47 \( 1 - 1.65T + 47T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + 6.24T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 - 9.31iT - 67T^{2} \)
71 \( 1 + 0.343T + 71T^{2} \)
73 \( 1 + 3.17T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 - 5.31T + 97T^{2} \)
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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

−10.73407305884765528279206925216, −10.24902351613322057564825658057, −9.083342638646280770869906167954, −8.537079843534201443580321171218, −7.23002652534497141722947204136, −6.35407543575744745154962984332, −5.53793252578151470096807336824, −4.07590662791715287407549147156, −3.39169834508120182371077619166, −2.28729350308410129217225848829, 0.810101194556645063547999786689, 1.78610389225483470104718565804, 3.52871098273680344426740647452, 4.62243800885960709008629911594, 5.78277286514227101071816200424, 6.51197269289063589164076919353, 7.85247088717976281955677758442, 8.218413224884172381167507537982, 9.179894096290314176672868295764, 9.893760986187044794616736145966

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