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  :  $\chi_{1008} (271, \cdot )$
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

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

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