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Properties

Label 2-1008-28.23-c0-0-1
Degree $2$
Conductor $1008$
Sign $0.895 + 0.444i$
Analytic cond. $0.503057$
Root an. cond. $0.709265$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)7-s + 13-s + (−1.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + (1.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (−1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (1.5 + 0.866i)79-s + (0.5 − 0.866i)91-s − 2·97-s + (−1.5 − 0.866i)103-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)7-s + 13-s + (−1.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + (1.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + 1.73i·43-s + (−0.499 − 0.866i)49-s + (−1 + 1.73i)61-s + (−1.5 + 0.866i)67-s + (−0.5 − 0.866i)73-s + (1.5 + 0.866i)79-s + (0.5 − 0.866i)91-s − 2·97-s + (−1.5 − 0.866i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.503057\)
Root analytic conductor: \(0.709265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.089836250\)
\(L(\frac12)\) \(\approx\) \(1.089836250\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−10.24510717110406885069276913112, −9.207418794365086457917037546001, −8.426468370268175024764815754304, −7.65259079006108557485197483987, −6.71531333641253673567420261415, −5.94398593579786385967362859058, −4.64540335931995716799197097869, −4.05197065307616256362195188110, −2.72902753207584850539372492885, −1.24909024554417592940450729484, 1.64777075275712270920995495399, 2.82279822552738431504329786803, 4.08133127446727830685973476746, 5.00343132322739890688605824133, 6.08562677816785593985410502332, 6.60628435969078377498052152443, 8.112762697373264439384570786860, 8.405080512342872457790042388779, 9.261681444660892454132230085584, 10.42155080946776623291331768250

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