Properties

Degree $2$
Conductor $1008$
Sign $0.00922 - 0.999i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.41i)3-s + (0.724 − 1.25i)5-s + (−0.5 − 0.866i)7-s + (−1.00 − 2.82i)9-s + (1 + 1.73i)11-s + (−2.44 + 4.24i)13-s + (1.05 + 2.28i)15-s + 2·17-s − 2.55·19-s + (1.72 + 0.158i)21-s + (−0.5 + 0.866i)23-s + (1.44 + 2.51i)25-s + (5.00 + 1.41i)27-s + (3.44 + 5.97i)29-s + (3 − 5.19i)31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + (0.324 − 0.561i)5-s + (−0.188 − 0.327i)7-s + (−0.333 − 0.942i)9-s + (0.301 + 0.522i)11-s + (−0.679 + 1.17i)13-s + (0.271 + 0.588i)15-s + 0.485·17-s − 0.585·19-s + (0.376 + 0.0346i)21-s + (−0.104 + 0.180i)23-s + (0.289 + 0.502i)25-s + (0.962 + 0.272i)27-s + (0.640 + 1.10i)29-s + (0.538 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00922 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00922 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.00922 - 0.999i$
Motivic weight: \(1\)
Character: $\chi_{1008} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.00922 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.124138752\)
\(L(\frac12)\) \(\approx\) \(1.124138752\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.724 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.44 - 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2.55T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.44 - 5.97i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.7T + 37T^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.44 - 5.97i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.27 + 5.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.44 - 11.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.101T + 71T^{2} \)
73 \( 1 + 6.89T + 73T^{2} \)
79 \( 1 + (0.949 + 1.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 16.8T + 89T^{2} \)
97 \( 1 + (1.44 + 2.51i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.883685567065142100191570738768, −9.591954884081203779428225928537, −8.821767651070754941225395021555, −7.61861189371329348064761192178, −6.59346717162850579785892586816, −5.88901167290752387832275338558, −4.58615775155479881587904839052, −4.44962215031996023637117947626, −2.96158404149998557851368046115, −1.31698624389686944890217139504, 0.60164079668848799712035087560, 2.25013061593942333521244211182, 3.08851703960304538358922525791, 4.67130306694046274869808185284, 5.76642568124034498705433522459, 6.23042405133499405351376482239, 7.15958967631356796316169491940, 7.991579957621757819259544721453, 8.761235067579355374281873258742, 10.09396562082272593694789489336

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