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  + (1.70 − 0.292i)3-s − 0.585i·5-s i·7-s + (2.82 − i)9-s + 2·11-s + 0.585·13-s + (−0.171 − i)15-s + 0.828i·17-s − 2.24i·19-s + (−0.292 − 1.70i)21-s − 4·23-s + 4.65·25-s + (4.53 − 2.53i)27-s + 0.828i·29-s − 6.82i·31-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)3-s − 0.261i·5-s − 0.377i·7-s + (0.942 − 0.333i)9-s + 0.603·11-s + 0.162·13-s + (−0.0442 − 0.258i)15-s + 0.200i·17-s − 0.514i·19-s + (−0.0639 − 0.372i)21-s − 0.834·23-s + 0.931·25-s + (0.872 − 0.487i)27-s + 0.153i·29-s − 1.22i·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  =  \((2,\ 672,\ (\ :1/2),\ 0.816 + 0.577i)\)
\(L(1)\)  \(\approx\)  \(2.08677 - 0.663255i\)
\(L(\frac12)\)  \(\approx\)  \(2.08677 - 0.663255i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.70 + 0.292i)T \)
7 \( 1 + iT \)
good5 \( 1 + 0.585iT - 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + 2.24iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 - 4.82T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 + 6.48iT - 43T^{2} \)
47 \( 1 + 9.65T + 47T^{2} \)
53 \( 1 - 9.31iT - 53T^{2} \)
59 \( 1 - 2.24T + 59T^{2} \)
61 \( 1 - 5.75T + 61T^{2} \)
67 \( 1 - 13.3iT - 67T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + 8.82T + 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 - 8.58T + 83T^{2} \)
89 \( 1 - 3.65iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−10.20213096196717457851579481657, −9.468537415113072962386151713309, −8.677465738646588709369434594774, −7.911015419463062002763321152741, −7.02056175251060120140440101471, −6.11411524484603789493411544650, −4.62350095383187703664797220501, −3.82360681883261776225163036186, −2.63341935216332842441543841242, −1.25594799337853660265011980218, 1.69348668093072361580232521910, 2.93753438758147736786661562843, 3.86415380415872231759763070043, 4.96607079726573133850491214520, 6.26809706881516065164723057622, 7.15114864368907221882507410314, 8.157285776247721789767632529271, 8.817665526844521624062573424260, 9.668108334542608589196679667267, 10.38769057845321714573139164857

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