Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.832 + 0.553i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)5-s + (−2.5 + 0.866i)7-s + (4.5 + 2.59i)11-s − 6.92i·13-s + (−3 − 1.73i)17-s + (1 + 1.73i)19-s + (6 − 3.46i)23-s + (−1 + 1.73i)25-s + 9·29-s + (0.5 − 0.866i)31-s + (−3 + 3.46i)35-s + (1 + 1.73i)37-s − 3.46i·41-s − 3.46i·43-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (0.670 − 0.387i)5-s + (−0.944 + 0.327i)7-s + (1.35 + 0.783i)11-s − 1.92i·13-s + (−0.727 − 0.420i)17-s + (0.229 + 0.397i)19-s + (1.25 − 0.722i)23-s + (−0.200 + 0.346i)25-s + 1.67·29-s + (0.0898 − 0.155i)31-s + (−0.507 + 0.585i)35-s + (0.164 + 0.284i)37-s − 0.541i·41-s − 0.528i·43-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.832 + 0.553i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (703, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.832 + 0.553i)\)
\(L(1)\)  \(\approx\)  \(1.728037418\)
\(L(\frac12)\)  \(\approx\)  \(1.728037418\)
\(L(\frac{3}{2})\)   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 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + (3 + 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6 + 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + (-9 + 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−9.894861910460469390555412424077, −9.105066421638555384748676992002, −8.470803708734452887368954340046, −7.18366102472951024186826040104, −6.48071264477018863662709027190, −5.62146624535533189997124188178, −4.73582601770195859028382620820, −3.46328607007613271714874120677, −2.47121977856574694614672051821, −0.933950935034233252038729125675, 1.30092379500974649699960088016, 2.67992477069804157496770614128, 3.77057343027783696062868461053, 4.64618146519514869628386735818, 6.30164219622437506199627978794, 6.38573516368759475917566712732, 7.21361517655575084220365219986, 8.779248243319662642077109801314, 9.181383348129956121802864640353, 9.888813602463832206969879485326

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