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Properties

Label 2-1008-7.2-c1-0-18
Degree $2$
Conductor $1008$
Sign $-0.968 + 0.250i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)5-s + (−2.5 − 0.866i)7-s + (−1.5 − 2.59i)11-s − 4·13-s + (−2 + 3.46i)19-s + (−2 − 3.46i)25-s − 9·29-s + (−0.5 − 0.866i)31-s + (−6 + 5.19i)35-s + (−4 + 6.92i)37-s + 10·43-s + (3 − 5.19i)47-s + (5.5 + 4.33i)49-s + (−1.5 − 2.59i)53-s − 9·55-s + ⋯
L(s)  = 1  + (0.670 − 1.16i)5-s + (−0.944 − 0.327i)7-s + (−0.452 − 0.783i)11-s − 1.10·13-s + (−0.458 + 0.794i)19-s + (−0.400 − 0.692i)25-s − 1.67·29-s + (−0.0898 − 0.155i)31-s + (−1.01 + 0.878i)35-s + (−0.657 + 1.13i)37-s + 1.52·43-s + (0.437 − 0.757i)47-s + (0.785 + 0.618i)49-s + (−0.206 − 0.356i)53-s − 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6930428719\)
\(L(\frac12)\) \(\approx\) \(0.6930428719\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.583423444422075426147263194586, −8.903813042686690022795908146808, −8.000527751493954204057624736319, −7.05787361980563476625985907781, −5.94054464528947039687856472998, −5.38543701349227714677832763856, −4.31276540220881250506272597446, −3.17173744395895041352264928028, −1.85551153601916422266158864366, −0.28727388955598231549747525318, 2.27293534115641032814031847579, 2.75429310240048749465030159754, 4.06905774142600557628826613199, 5.35885184168925410594879647393, 6.12120921839991129345018684980, 7.13677126661684041108292173676, 7.40168535092100638650258680730, 9.026726245284048765918536737270, 9.552288013647611808113491293318, 10.34155434247508071491124503697

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