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Properties

Label 2-1008-7.2-c1-0-16
Degree $2$
Conductor $1008$
Sign $-0.0158 + 0.999i$
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  + (1.63 − 2.83i)5-s + (1.5 − 2.17i)7-s + (−1.63 − 2.83i)11-s + 6.27·13-s + (−2 − 3.46i)17-s + (−3.13 + 5.43i)19-s + (−2 + 3.46i)23-s + (−2.86 − 4.95i)25-s − 5.27·29-s + (−0.5 − 0.866i)31-s + (−3.72 − 7.82i)35-s + (1.13 − 1.97i)37-s + 4.54·41-s − 0.274·43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (0.732 − 1.26i)5-s + (0.566 − 0.823i)7-s + (−0.493 − 0.855i)11-s + 1.74·13-s + (−0.485 − 0.840i)17-s + (−0.719 + 1.24i)19-s + (−0.417 + 0.722i)23-s + (−0.572 − 0.991i)25-s − 0.979·29-s + (−0.0898 − 0.155i)31-s + (−0.629 − 1.32i)35-s + (0.186 − 0.323i)37-s + 0.710·41-s − 0.0419·43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0158 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.0158 + 0.999i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.0158 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.864721144\)
\(L(\frac12)\) \(\approx\) \(1.864721144\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.17i)T \)
good5 \( 1 + (-1.63 + 2.83i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.63 + 2.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.13 - 5.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.13 + 1.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 0.274T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.63 - 8.03i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.637 - 1.10i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.137 + 0.238i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-2.13 - 3.70i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.77 + 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + (5.27 - 9.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−9.670817932816418326899635502622, −8.795784765018193096975017033850, −8.306204915797760913707319384898, −7.39424007239327348974436667153, −5.97775941958449552270528701297, −5.61378896072307341938362917863, −4.43496002518057445278987340040, −3.63369878699257650002087708847, −1.87454138836303077871324883769, −0.889335587118196056057092278722, 1.90054759399133256678471167663, 2.58021730348009030879718591430, 3.87942952118571722386215474297, 5.05496716813524504857628313428, 6.19265472637102351053162063742, 6.48670646447873626427461200712, 7.67739631790014873105662738771, 8.595664943169814163266077976555, 9.297453781534938858641044722593, 10.39622780421295588684095358555

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