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Properties

Label 2-1008-48.35-c1-0-16
Degree $2$
Conductor $1008$
Sign $0.903 - 0.429i$
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  + (−0.351 − 1.36i)2-s + (−1.75 + 0.964i)4-s + (2.96 + 2.96i)5-s + 7-s + (1.93 + 2.06i)8-s + (3.02 − 5.11i)10-s + (−0.569 + 0.569i)11-s + (2.53 + 2.53i)13-s + (−0.351 − 1.36i)14-s + (2.14 − 3.37i)16-s − 1.03i·17-s + (−5.23 + 5.23i)19-s + (−8.06 − 2.33i)20-s + (0.981 + 0.579i)22-s − 8.71i·23-s + ⋯
L(s)  = 1  + (−0.248 − 0.968i)2-s + (−0.876 + 0.482i)4-s + (1.32 + 1.32i)5-s + 0.377·7-s + (0.684 + 0.728i)8-s + (0.955 − 1.61i)10-s + (−0.171 + 0.171i)11-s + (0.703 + 0.703i)13-s + (−0.0940 − 0.366i)14-s + (0.535 − 0.844i)16-s − 0.251i·17-s + (−1.20 + 1.20i)19-s + (−1.80 − 0.523i)20-s + (0.209 + 0.123i)22-s − 1.81i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.903 - 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.597536354\)
\(L(\frac12)\) \(\approx\) \(1.597536354\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.351 + 1.36i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (-2.96 - 2.96i)T + 5iT^{2} \)
11 \( 1 + (0.569 - 0.569i)T - 11iT^{2} \)
13 \( 1 + (-2.53 - 2.53i)T + 13iT^{2} \)
17 \( 1 + 1.03iT - 17T^{2} \)
19 \( 1 + (5.23 - 5.23i)T - 19iT^{2} \)
23 \( 1 + 8.71iT - 23T^{2} \)
29 \( 1 + (6.05 - 6.05i)T - 29iT^{2} \)
31 \( 1 - 3.00iT - 31T^{2} \)
37 \( 1 + (0.149 - 0.149i)T - 37iT^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + (-1.73 - 1.73i)T + 43iT^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 + (6.04 + 6.04i)T + 53iT^{2} \)
59 \( 1 + (6.05 - 6.05i)T - 59iT^{2} \)
61 \( 1 + (-5.81 - 5.81i)T + 61iT^{2} \)
67 \( 1 + (-0.0256 + 0.0256i)T - 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 5.38iT - 73T^{2} \)
79 \( 1 - 3.89iT - 79T^{2} \)
83 \( 1 + (1.40 + 1.40i)T + 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−10.37167432588317665445819463284, −9.328513887301982703515854393439, −8.709045560343583032613584497700, −7.57914880479149097634210067850, −6.55502474608907435457988448514, −5.80637650871941103878977712654, −4.56830205852268642520861036756, −3.46268060455060381330390198635, −2.37363812207659045673239269807, −1.69539519740523260694357167016, 0.832949950820463100490983277654, 2.03993650208376135770043490711, 4.04684591515049446083089061698, 4.98749306770190616418521459840, 5.74560770692162666605519933219, 6.18066227616264469455544247416, 7.56612518541817125197498964512, 8.292359947982764861296304702886, 9.109248520403177664854719095381, 9.479593295809940893852954862682

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