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Properties

Label 2-1008-21.17-c1-0-7
Degree $2$
Conductor $1008$
Sign $0.168 - 0.985i$
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  + (2.09 + 3.62i)5-s + (1.62 + 2.09i)7-s + (2.59 + 1.5i)11-s − 2.44i·13-s + (−0.507 + 0.878i)17-s + (0.878 − 0.507i)19-s + (3.67 − 2.12i)23-s + (−6.24 + 10.8i)25-s − 1.24i·29-s + (−4.86 − 2.80i)31-s + (−4.18 + 10.2i)35-s + (−4.12 − 7.13i)37-s + 2.02·41-s − 8.24·43-s + (0.507 + 0.878i)47-s + ⋯
L(s)  = 1  + (0.935 + 1.61i)5-s + (0.612 + 0.790i)7-s + (0.783 + 0.452i)11-s − 0.679i·13-s + (−0.123 + 0.213i)17-s + (0.201 − 0.116i)19-s + (0.766 − 0.442i)23-s + (−1.24 + 2.16i)25-s − 0.230i·29-s + (−0.873 − 0.504i)31-s + (−0.706 + 1.73i)35-s + (−0.677 − 1.17i)37-s + 0.316·41-s − 1.25·43-s + (0.0739 + 0.128i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.168 - 0.985i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.168 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.069383237\)
\(L(\frac12)\) \(\approx\) \(2.069383237\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
good5 \( 1 + (-2.09 - 3.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.878 + 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.67 + 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.24iT - 29T^{2} \)
31 \( 1 + (4.86 + 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.12 + 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (-0.507 - 0.878i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.07 - 0.621i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.76 + 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 - 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.62 + 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.76iT - 97T^{2} \)
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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

−10.13956023448837574019595045952, −9.459271655557506711624553540945, −8.587849648663850400908098381263, −7.43970725803390592415027321353, −6.75898156283073763165416656913, −5.92338233099151805691132945182, −5.17882138537958431721563387490, −3.70567734099486976384918283831, −2.64813632942778248163509947592, −1.83168900885703679754039013027, 1.05064315691622734939393738901, 1.79591602732710900154590549549, 3.64749594222050520309198451140, 4.70309817167901008740532944520, 5.23203400396574545207754618534, 6.31004615083620550137648153659, 7.25111696833699606411365189859, 8.406906099513226626161571374765, 8.935190081598548881330113736956, 9.609119064414538728171807297627

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