Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 7 $
Sign $-0.987 + 0.157i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.434 + 1.67i)3-s + (−1.21 + 2.10i)5-s + (0.5 + 0.866i)7-s + (−2.62 + 1.45i)9-s + (0.379 + 0.657i)11-s + (−1.11 + 1.92i)13-s + (−4.06 − 1.12i)15-s − 7.04·17-s + 1.37·19-s + (−1.23 + 1.21i)21-s + (3.51 − 6.09i)23-s + (−0.467 − 0.810i)25-s + (−3.58 − 3.76i)27-s + (−0.418 − 0.724i)29-s + (−0.265 + 0.459i)31-s + ⋯
L(s)  = 1  + (0.250 + 0.967i)3-s + (−0.544 + 0.943i)5-s + (0.188 + 0.327i)7-s + (−0.874 + 0.485i)9-s + (0.114 + 0.198i)11-s + (−0.309 + 0.535i)13-s + (−1.05 − 0.290i)15-s − 1.70·17-s + 0.314·19-s + (−0.269 + 0.265i)21-s + (0.733 − 1.27i)23-s + (−0.0935 − 0.162i)25-s + (−0.689 − 0.724i)27-s + (−0.0776 − 0.134i)29-s + (−0.0476 + 0.0825i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.987 + 0.157i$
motivic weight  =  \(1\)
character  :  $\chi_{1008} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1008,\ (\ :1/2),\ -0.987 + 0.157i)\)
\(L(1)\)  \(\approx\)  \(0.9487134374\)
\(L(\frac12)\)  \(\approx\)  \(0.9487134374\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.434 - 1.67i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.21 - 2.10i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.379 - 0.657i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.11 - 1.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.04T + 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
23 \( 1 + (-3.51 + 6.09i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.418 + 0.724i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.265 - 0.459i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 + (4.42 - 7.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.70 - 6.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.39 + 5.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.607T + 53T^{2} \)
59 \( 1 + (0.581 - 1.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.85 + 10.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.152 - 0.264i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + (-7.62 - 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.18 - 14.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.63T + 89T^{2} \)
97 \( 1 + (-5.46 - 9.46i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46142730527820498228390124493, −9.543540782384529733796899270756, −8.855243971259693496672113127052, −8.038006338218803729445675376995, −6.97734752784567410956640144998, −6.27015608863450345683217992854, −4.87164922039547150671891256397, −4.31002218442586362228626244198, −3.14987641084811615120284024332, −2.32398024122800783458981058269, 0.40871027364755789465793851418, 1.65742367255365513695835255093, 3.00679650523693520568243175533, 4.20392260048891755972278103661, 5.16853814246002529630386365425, 6.18197342284771322994396476761, 7.28251249962233822793672769634, 7.71305721484478116321361779284, 8.873484886393490395516266413554, 8.983993115645955111537675325189

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