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Properties

Label 2-1008-63.47-c1-0-6
Degree $2$
Conductor $1008$
Sign $0.0614 - 0.998i$
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  + (−1.33 − 1.10i)3-s − 0.0676·5-s + (2.64 − 0.142i)7-s + (0.554 + 2.94i)9-s + 3.92i·11-s + (−3.32 + 1.92i)13-s + (0.0901 + 0.0747i)15-s + (−0.775 − 1.34i)17-s + (−5.06 − 2.92i)19-s + (−3.67 − 2.73i)21-s + 5.52i·23-s − 4.99·25-s + (2.52 − 4.54i)27-s + (1.20 + 0.697i)29-s + (−1.09 − 0.632i)31-s + ⋯
L(s)  = 1  + (−0.769 − 0.638i)3-s − 0.0302·5-s + (0.998 − 0.0538i)7-s + (0.184 + 0.982i)9-s + 1.18i·11-s + (−0.922 + 0.532i)13-s + (0.0232 + 0.0193i)15-s + (−0.188 − 0.325i)17-s + (−1.16 − 0.670i)19-s + (−0.802 − 0.596i)21-s + 1.15i·23-s − 0.999·25-s + (0.485 − 0.874i)27-s + (0.224 + 0.129i)29-s + (−0.196 − 0.113i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0614 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.0614 - 0.998i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.0614 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7908855677\)
\(L(\frac12)\) \(\approx\) \(0.7908855677\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.10i)T \)
7 \( 1 + (-2.64 + 0.142i)T \)
good5 \( 1 + 0.0676T + 5T^{2} \)
11 \( 1 - 3.92iT - 11T^{2} \)
13 \( 1 + (3.32 - 1.92i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.775 + 1.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.06 + 2.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.52iT - 23T^{2} \)
29 \( 1 + (-1.20 - 0.697i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.09 + 0.632i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.35 - 7.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.17 - 8.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.735 - 1.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.77 - 3.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.28 + 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.70 - 8.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0705 + 0.0407i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.67 - 13.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.30iT - 71T^{2} \)
73 \( 1 + (-6.12 + 3.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.42 + 5.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.93 + 6.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.84 - 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.363 - 0.209i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.24194658091037190022032312139, −9.443899279440264791916002384051, −8.305558859744558111004344548689, −7.43183454627138959873682529935, −6.99156605344460878072391365743, −5.89336936896134596964502783824, −4.81777433474584738018151555459, −4.42398981646540851942794403915, −2.39704831178248623151733206339, −1.54940154682292819554105158155, 0.39351911477433075571870285987, 2.17612897720076527735205360514, 3.70468273803819480515008981185, 4.50739250855118497925897005975, 5.50198900261242795083476725287, 6.05166215988889410896953622494, 7.22129852684413699663274403208, 8.270552847157342723264931798364, 8.854611915544535277920262958475, 9.985926671074871150222395868595

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