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Properties

Label 2-1008-7.4-c1-0-3
Degree $2$
Conductor $1008$
Sign $0.386 - 0.922i$
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 + (0.5 + 2.59i)7-s + (−1.5 + 2.59i)11-s + 2·13-s + (−3 + 5.19i)17-s + (1 + 1.73i)19-s + (−3 − 5.19i)23-s + (−2 + 3.46i)25-s + 9·29-s + (−3.5 + 6.06i)31-s + (6 − 5.19i)35-s + (5 + 8.66i)37-s + 4·43-s + (6 + 10.3i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + (0.188 + 0.981i)7-s + (−0.452 + 0.783i)11-s + 0.554·13-s + (−0.727 + 1.26i)17-s + (0.229 + 0.397i)19-s + (−0.625 − 1.08i)23-s + (−0.400 + 0.692i)25-s + 1.67·29-s + (−0.628 + 1.08i)31-s + (1.01 − 0.878i)35-s + (0.821 + 1.42i)37-s + 0.609·43-s + (0.875 + 1.51i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.126053555\)
\(L(\frac12)\) \(\approx\) \(1.126053555\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
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 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-6 - 10.3i)T + (-23.5 + 40.7i)T^{2} \)
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 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−10.10808856957645975351917607242, −9.050090566051939734338331164876, −8.373558269339540129128116687511, −8.044268050005006800554241031652, −6.65014892356314452493807033363, −5.76847484041962998092619467125, −4.73090435828397489172979158510, −4.18710654418748340735998194751, −2.66931115540000292494786191131, −1.37683711622456274284589108222, 0.55254619880897760023605978024, 2.52633180207511502028265536695, 3.50091624768140207978449546864, 4.28473550331234860402312293648, 5.55560373656079630959494587094, 6.62070536059120507158464048755, 7.35038052014407444465084794476, 7.87791013055404234309224584781, 8.970626824856214767523167828131, 9.975646504346794274877075024258

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