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Properties

Label 2-1008-7.4-c1-0-16
Degree $2$
Conductor $1008$
Sign $-0.701 + 0.712i$
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  + (0.5 − 2.59i)7-s − 7·13-s + (−3.5 − 6.06i)19-s + (2.5 − 4.33i)25-s + (−3.5 + 6.06i)31-s + (0.5 + 0.866i)37-s − 5·43-s + (−6.5 − 2.59i)49-s + (−7 − 12.1i)61-s + (5.5 − 9.52i)67-s + (3.5 − 6.06i)73-s + (−6.5 − 11.2i)79-s + (−3.5 + 18.1i)91-s + 14·97-s + (−3.5 − 6.06i)103-s + ⋯
L(s)  = 1  + (0.188 − 0.981i)7-s − 1.94·13-s + (−0.802 − 1.39i)19-s + (0.5 − 0.866i)25-s + (−0.628 + 1.08i)31-s + (0.0821 + 0.142i)37-s − 0.762·43-s + (−0.928 − 0.371i)49-s + (−0.896 − 1.55i)61-s + (0.671 − 1.16i)67-s + (0.409 − 0.709i)73-s + (−0.731 − 1.26i)79-s + (−0.366 + 1.90i)91-s + 1.42·97-s + (−0.344 − 0.597i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7997956491\)
\(L(\frac12)\) \(\approx\) \(0.7997956491\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 + 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−9.739964533280980824734511381197, −8.876800781619641351557387430949, −7.86380392439374857471887658001, −7.11426637292004039037432808600, −6.50175722611677744829777055765, −4.93238082688137468244823726078, −4.62166371418805977949109790621, −3.23044867448268840131880782797, −2.08520216175623550365198959326, −0.33912932270619432428170501962, 1.88901651232010319783404348713, 2.80204972038659717137231396963, 4.14987164623491398145971295562, 5.18893701315880067251559790016, 5.84779876966750141570473531862, 6.99022875816697026822061597234, 7.79694550440379629369947205093, 8.625070940816977595190755931455, 9.522503565607493123660424770732, 10.09483885543316872585529127791

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