Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.0633 + 0.997i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.35 + 1.93i)5-s + (−2.5 + 0.866i)7-s + (3.35 + 1.93i)11-s − 3.46i·13-s + (−2 − 3.46i)19-s + (−6.70 + 3.87i)23-s + (5.00 − 8.66i)25-s + 6.70·29-s + (0.5 − 0.866i)31-s + (6.70 − 7.74i)35-s + (−2 − 3.46i)37-s − 7.74i·41-s − 6.92i·43-s + (−6.70 − 11.6i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (−1.50 + 0.866i)5-s + (−0.944 + 0.327i)7-s + (1.01 + 0.583i)11-s − 0.960i·13-s + (−0.458 − 0.794i)19-s + (−1.39 + 0.807i)23-s + (1.00 − 1.73i)25-s + 1.24·29-s + (0.0898 − 0.155i)31-s + (1.13 − 1.30i)35-s + (−0.328 − 0.569i)37-s − 1.20i·41-s − 1.05i·43-s + (−0.978 − 1.69i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0633 + 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (703, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.0633 + 0.997i)\)
\(L(1)\)  \(\approx\)  \(0.4555432341\)
\(L(\frac12)\)  \(\approx\)  \(0.4555432341\)
\(L(\frac{3}{2})\)   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 \( 1 + (3.35 - 1.93i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.35 - 1.93i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.70 - 3.87i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.74iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 + (6.70 + 11.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.35 + 5.80i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.35 - 5.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9 + 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 3.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + (6.70 - 3.87i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−9.932123542239718294978478186236, −8.799663945491098861240055783144, −8.072182112075953068794277111720, −7.02422873041533328337385869638, −6.70728574643668517469699834824, −5.48139927121624473964138070826, −4.04772475234453435001718670517, −3.57417402299498779082517245199, −2.46177971153893970063453403682, −0.23574733024820085158381789164, 1.20503153995458890935588596464, 3.16055741595749354742153670064, 4.12452520169172423709139938571, 4.50561423919173909107225031819, 6.17443967569189105077382318803, 6.69079304558155988333959294635, 7.898251038508377305355842064333, 8.433193186912317092813652821737, 9.251356532149790697729744734655, 10.08273138447760577921052761749

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