Properties

Degree $2$
Conductor $1008$
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

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.987 - 0.157i$
Motivic weight: \(1\)
Character: $\chi_{1008} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.987 - 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9487134374\)
\(L(\frac12)\) \(\approx\) \(0.9487134374\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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