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Properties

Label 2-1008-252.103-c1-0-33
Degree $2$
Conductor $1008$
Sign $0.454 + 0.890i$
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  + (0.284 + 1.70i)3-s + (−0.950 + 0.548i)5-s + (−1.53 − 2.15i)7-s + (−2.83 + 0.972i)9-s + (−0.0872 − 0.0503i)11-s + (−4.27 − 2.46i)13-s + (−1.20 − 1.46i)15-s + (3.23 − 1.86i)17-s + (2.54 − 4.41i)19-s + (3.25 − 3.23i)21-s + (7.97 − 4.60i)23-s + (−1.89 + 3.28i)25-s + (−2.46 − 4.57i)27-s + (−3.96 − 6.86i)29-s + 4.83·31-s + ⋯
L(s)  = 1  + (0.164 + 0.986i)3-s + (−0.425 + 0.245i)5-s + (−0.578 − 0.815i)7-s + (−0.945 + 0.324i)9-s + (−0.0263 − 0.0151i)11-s + (−1.18 − 0.683i)13-s + (−0.311 − 0.378i)15-s + (0.784 − 0.453i)17-s + (0.584 − 1.01i)19-s + (0.709 − 0.704i)21-s + (1.66 − 0.959i)23-s + (−0.379 + 0.657i)25-s + (−0.475 − 0.879i)27-s + (−0.735 − 1.27i)29-s + 0.868·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.454 + 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.454 + 0.890i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.454 + 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8842405735\)
\(L(\frac12)\) \(\approx\) \(0.8842405735\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.284 - 1.70i)T \)
7 \( 1 + (1.53 + 2.15i)T \)
good5 \( 1 + (0.950 - 0.548i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.0872 + 0.0503i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.27 + 2.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.23 + 1.86i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.54 + 4.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.97 + 4.60i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.96 + 6.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.83T + 31T^{2} \)
37 \( 1 + (2.77 - 4.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.91 - 2.25i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 - 1.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.35T + 47T^{2} \)
53 \( 1 + (6.53 + 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 2.88T + 59T^{2} \)
61 \( 1 - 9.48iT - 61T^{2} \)
67 \( 1 + 12.5iT - 67T^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + (-5.92 + 3.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.67iT - 79T^{2} \)
83 \( 1 + (-0.112 - 0.195i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.90 + 2.83i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.47 - 3.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.724393006377263126881665165028, −9.368094126910994928319768414754, −8.054710691487346904735857368642, −7.43392810787580506474058998433, −6.49071396141909236705387267554, −5.16014559769395224843556430956, −4.60848555828807531738255503263, −3.34636300370085961693704286452, −2.84680128545855702366589917722, −0.40467024694233478109566498635, 1.43479448438596563156096532846, 2.68422484730038753408442810216, 3.60811071523229802433698245603, 5.12853101539621359050772168547, 5.84623974059461792689288572673, 6.91294907727170016550698173144, 7.52907061214485578539879016861, 8.385206130264459717536993159279, 9.219224433331301791509295598555, 9.858712191314387026308682287527

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