Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{2} \cdot 7 $
Sign $0.121 - 0.992i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 − 0.756i)2-s + (0.856 + 1.80i)4-s + 4.10i·5-s + 7-s + (0.343 − 2.80i)8-s + (3.10 − 4.90i)10-s − 2.67i·11-s + 3.02i·13-s + (−1.19 − 0.756i)14-s + (−2.53 + 3.09i)16-s + 5.12·17-s + 2.78i·19-s + (−7.41 + 3.51i)20-s + (−2.02 + 3.19i)22-s − 7.12·23-s + ⋯
L(s)  = 1  + (−0.845 − 0.534i)2-s + (0.428 + 0.903i)4-s + 1.83i·5-s + 0.377·7-s + (0.121 − 0.992i)8-s + (0.981 − 1.55i)10-s − 0.807i·11-s + 0.838i·13-s + (−0.319 − 0.202i)14-s + (−0.633 + 0.773i)16-s + 1.24·17-s + 0.637i·19-s + (−1.65 + 0.785i)20-s + (−0.431 + 0.682i)22-s − 1.48·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.121 - 0.992i$
motivic weight  =  \(1\)
character  :  $\chi_{504} (253, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 504,\ (\ :1/2),\ 0.121 - 0.992i)$
$L(1)$  $\approx$  $0.652325 + 0.577334i$
$L(\frac12)$  $\approx$  $0.652325 + 0.577334i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 0.756i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 4.10iT - 5T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 - 3.02iT - 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + 7.12T + 23T^{2} \)
29 \( 1 - 8.83iT - 29T^{2} \)
31 \( 1 + 1.42T + 31T^{2} \)
37 \( 1 - 1.42iT - 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 - 2.39iT - 43T^{2} \)
47 \( 1 - 9.56T + 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 + 5.17iT - 61T^{2} \)
67 \( 1 + 0.244iT - 67T^{2} \)
71 \( 1 - 4.27T + 71T^{2} \)
73 \( 1 + 4.15T + 73T^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 - 9.35iT - 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 6.69T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.95623409057398416159455216725, −10.35149481568703580864662485693, −9.618406022181595106139587239462, −8.390563780803850017617230491217, −7.59509921348481940258036280068, −6.79394240745581040817273494402, −5.86126396549973120084691332221, −3.82460985056023563629016957962, −3.08878082193135043807338288817, −1.82482043756576074841121446796, 0.69810166096243706579541683139, 1.98894732046040587315788481101, 4.27696268260208546521098781286, 5.26249218496061799532573904290, 5.90337893711496655524306614090, 7.51723939462532092463076526604, 8.044330232177693814165179140587, 8.832118761155878983713157536174, 9.718559286654454627348350139245, 10.28533231332801297061159328885

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