Properties

Degree 2
Conductor $ 2^{5} \cdot 3 \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $-0.816 + 0.577i$
motivic weight  =  \(1\)
character  :  $\chi_{672} (575, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 672,\ (\ :1/2),\ -0.816 + 0.577i)\)
\(L(1)\)  \(\approx\)  \(0.448578 - 1.41134i\)
\(L(\frac12)\)  \(\approx\)  \(0.448578 - 1.41134i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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