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Properties

Label 2-1008-48.35-c1-0-3
Degree $2$
Conductor $1008$
Sign $-0.999 + 0.0183i$
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  + (1.13 + 0.844i)2-s + (0.572 + 1.91i)4-s + (−2.12 − 2.12i)5-s + 7-s + (−0.969 + 2.65i)8-s + (−0.613 − 4.19i)10-s + (−4.03 + 4.03i)11-s + (−4.91 − 4.91i)13-s + (1.13 + 0.844i)14-s + (−3.34 + 2.19i)16-s + 4.76i·17-s + (−2.21 + 2.21i)19-s + (2.84 − 5.27i)20-s + (−7.99 + 1.16i)22-s + 6.91i·23-s + ⋯
L(s)  = 1  + (0.802 + 0.597i)2-s + (0.286 + 0.958i)4-s + (−0.948 − 0.948i)5-s + 0.377·7-s + (−0.342 + 0.939i)8-s + (−0.194 − 1.32i)10-s + (−1.21 + 1.21i)11-s + (−1.36 − 1.36i)13-s + (0.303 + 0.225i)14-s + (−0.835 + 0.548i)16-s + 1.15i·17-s + (−0.508 + 0.508i)19-s + (0.636 − 1.18i)20-s + (−1.70 + 0.249i)22-s + 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.999 + 0.0183i$
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: \((2,\ 1008,\ (\ :1/2),\ -0.999 + 0.0183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7550998426\)
\(L(\frac12)\) \(\approx\) \(0.7550998426\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.844i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (2.12 + 2.12i)T + 5iT^{2} \)
11 \( 1 + (4.03 - 4.03i)T - 11iT^{2} \)
13 \( 1 + (4.91 + 4.91i)T + 13iT^{2} \)
17 \( 1 - 4.76iT - 17T^{2} \)
19 \( 1 + (2.21 - 2.21i)T - 19iT^{2} \)
23 \( 1 - 6.91iT - 23T^{2} \)
29 \( 1 + (2.61 - 2.61i)T - 29iT^{2} \)
31 \( 1 - 0.712iT - 31T^{2} \)
37 \( 1 + (-5.73 + 5.73i)T - 37iT^{2} \)
41 \( 1 - 3.59T + 41T^{2} \)
43 \( 1 + (-3.36 - 3.36i)T + 43iT^{2} \)
47 \( 1 + 6.04T + 47T^{2} \)
53 \( 1 + (4.53 + 4.53i)T + 53iT^{2} \)
59 \( 1 + (-4.82 + 4.82i)T - 59iT^{2} \)
61 \( 1 + (6.72 + 6.72i)T + 61iT^{2} \)
67 \( 1 + (3.87 - 3.87i)T - 67iT^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 - 4.61iT - 73T^{2} \)
79 \( 1 - 7.43iT - 79T^{2} \)
83 \( 1 + (2.44 + 2.44i)T + 83iT^{2} \)
89 \( 1 + 4.23T + 89T^{2} \)
97 \( 1 + 6.76T + 97T^{2} \)
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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

−10.46354147389702609374685200935, −9.461162445870927277153830608579, −8.179983905300095582803716645856, −7.82148211227797168660997346588, −7.34349426397274090545252058371, −5.81672759983036013546871934598, −5.05135379498913625953572322632, −4.50914887442610909046846806628, −3.44427788359108039699455445944, −2.11440508920124482182212842744, 0.24098782220511007771058373305, 2.46933754994054382685867275379, 2.91922392308457537938826129554, 4.27787895206514437361598235268, 4.84449928140173325526360766644, 6.05275548694538918120653441200, 7.00433724789473829172238070016, 7.62363729807155388828078100578, 8.783950266654337596379862255046, 9.839534786380446220765756763214

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