Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $0.913 - 0.407i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.60 − 0.649i)3-s + (−0.468 + 0.811i)5-s + (0.5 + 0.866i)7-s + (2.15 − 2.08i)9-s + (2.48 + 4.30i)11-s + (−0.622 + 1.07i)13-s + (−0.225 + 1.60i)15-s + 5.22·17-s − 5.18·19-s + (1.36 + 1.06i)21-s + (−1.00 + 1.73i)23-s + (2.06 + 3.57i)25-s + (2.10 − 4.74i)27-s + (−3.43 − 5.95i)29-s + (−2.86 + 4.96i)31-s + ⋯
L(s)  = 1  + (0.927 − 0.374i)3-s + (−0.209 + 0.362i)5-s + (0.188 + 0.327i)7-s + (0.718 − 0.695i)9-s + (0.749 + 1.29i)11-s + (−0.172 + 0.298i)13-s + (−0.0581 + 0.414i)15-s + 1.26·17-s − 1.18·19-s + (0.297 + 0.232i)21-s + (−0.209 + 0.362i)23-s + (0.412 + 0.714i)25-s + (0.405 − 0.913i)27-s + (−0.638 − 1.10i)29-s + (−0.514 + 0.891i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.913 - 0.407i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ 0.913 - 0.407i)\)
\(L(1)\)  \(\approx\)  \(2.277735950\)
\(L(\frac12)\)  \(\approx\)  \(2.277735950\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.60 + 0.649i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.468 - 0.811i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.48 - 4.30i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.622 - 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5.22T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + (1.00 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.43 + 5.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.86 - 4.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (-5.73 + 9.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.80 - 8.31i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.984 + 1.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.63T + 53T^{2} \)
59 \( 1 + (-2.43 + 4.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.52 - 2.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.573 - 0.994i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.83T + 71T^{2} \)
73 \( 1 - 6.10T + 73T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.431 + 0.747i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + (3.78 + 6.55i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.613313139476417931860186211483, −9.399968942385806483774155750637, −8.253253151281955180733714447938, −7.52362871823919186202933656638, −6.89791231897976564404374116807, −5.88860527578448593865401029384, −4.49166626005601476520030052765, −3.71681788162767728136194258438, −2.51417421035978497324770397266, −1.55429264845187736571633896404, 1.07576780015636935545019452205, 2.61862361815010886389404078914, 3.68596881766531208926260529681, 4.35090966548687044127653355724, 5.53119220913855343133653045655, 6.54876907001369758443615649599, 7.78184134377494251563916804782, 8.217121071004584759584339007414, 9.041810580685306115199939893101, 9.746954332143724002072719170298

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