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Properties

Label 2-1008-63.4-c1-0-20
Degree $2$
Conductor $1008$
Sign $0.181 + 0.983i$
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  + (−0.444 + 1.67i)3-s − 3.52·5-s + (−1.16 + 2.37i)7-s + (−2.60 − 1.48i)9-s + 2.32·11-s + (−2.35 + 4.08i)13-s + (1.56 − 5.90i)15-s + (−0.636 + 1.10i)17-s + (−2.78 − 4.82i)19-s + (−3.45 − 3.00i)21-s + 3.29·23-s + 7.45·25-s + (3.64 − 3.69i)27-s + (−4.32 − 7.48i)29-s + (4.25 + 7.37i)31-s + ⋯
L(s)  = 1  + (−0.256 + 0.966i)3-s − 1.57·5-s + (−0.441 + 0.897i)7-s + (−0.868 − 0.496i)9-s + 0.699·11-s + (−0.654 + 1.13i)13-s + (0.405 − 1.52i)15-s + (−0.154 + 0.267i)17-s + (−0.638 − 1.10i)19-s + (−0.754 − 0.656i)21-s + 0.687·23-s + 1.49·25-s + (0.702 − 0.711i)27-s + (−0.802 − 1.38i)29-s + (0.764 + 1.32i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.181 + 0.983i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1487701791\)
\(L(\frac12)\) \(\approx\) \(0.1487701791\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.444 - 1.67i)T \)
7 \( 1 + (1.16 - 2.37i)T \)
good5 \( 1 + 3.52T + 5T^{2} \)
11 \( 1 - 2.32T + 11T^{2} \)
13 \( 1 + (2.35 - 4.08i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.636 - 1.10i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.78 + 4.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.29T + 23T^{2} \)
29 \( 1 + (4.32 + 7.48i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.25 - 7.37i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.84 + 4.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.66 + 2.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0444 + 0.0769i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.52 + 6.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.41 + 5.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.99 + 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.67 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.06 - 5.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.30T + 71T^{2} \)
73 \( 1 + (-6.64 + 11.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.01 - 8.68i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.90 - 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.561 - 0.972i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.50 + 6.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.612136502564115740370452508791, −8.957233268627791701725659121684, −8.439821556828678393588421114262, −7.13196511104233196258814425048, −6.46956225950061112529339531090, −5.21127988705143997603048779108, −4.33606831538491621847451649676, −3.72530808977626043487468731685, −2.55621054489695452442060854274, −0.083048006438092292259076391904, 1.09922882939623470433381429145, 2.94481700405233721126573169316, 3.82654148079596888857063207880, 4.83528240924578457430526414205, 6.11553206575456775867743883954, 6.98392674530039397230425189031, 7.65993377898569064227053513408, 8.073664973106764973335667088834, 9.178257651643290475897060871994, 10.41837238734744660417606361239

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