Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.935 + 0.352i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 − 0.848i)2-s + (0.559 + 1.92i)4-s + 0.940i·5-s + 7-s + (0.996 − 2.64i)8-s + (0.798 − 1.06i)10-s + 5.98i·11-s − 6.59i·13-s + (−1.13 − 0.848i)14-s + (−3.37 + 2.14i)16-s + 2.64·17-s − 5.83i·19-s + (−1.80 + 0.526i)20-s + (5.08 − 6.77i)22-s − 2.88·23-s + ⋯
L(s)  = 1  + (−0.799 − 0.600i)2-s + (0.279 + 0.960i)4-s + 0.420i·5-s + 0.377·7-s + (0.352 − 0.935i)8-s + (0.252 − 0.336i)10-s + 1.80i·11-s − 1.82i·13-s + (−0.302 − 0.226i)14-s + (−0.843 + 0.537i)16-s + 0.640·17-s − 1.33i·19-s + (−0.403 + 0.117i)20-s + (1.08 − 1.44i)22-s − 0.601·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.935 + 0.352i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.935 + 0.352i)$
$L(1)$  $\approx$  $1.199951764$
$L(\frac12)$  $\approx$  $1.199951764$
$L(\frac{3}{2})$   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 \( 1 + (1.13 + 0.848i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 0.940iT - 5T^{2} \)
11 \( 1 - 5.98iT - 11T^{2} \)
13 \( 1 + 6.59iT - 13T^{2} \)
17 \( 1 - 2.64T + 17T^{2} \)
19 \( 1 + 5.83iT - 19T^{2} \)
23 \( 1 + 2.88T + 23T^{2} \)
29 \( 1 + 3.09iT - 29T^{2} \)
31 \( 1 - 3.52T + 31T^{2} \)
37 \( 1 + 0.213iT - 37T^{2} \)
41 \( 1 - 1.63T + 41T^{2} \)
43 \( 1 - 7.16iT - 43T^{2} \)
47 \( 1 - 9.32T + 47T^{2} \)
53 \( 1 - 7.51iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 - 1.48iT - 61T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 - 1.54T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 - 8.74iT - 83T^{2} \)
89 \( 1 + 7.50T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.665550675263673853721340622971, −8.679609744356778774825433142002, −7.73678039960585991182496513550, −7.41477768182220243414423488804, −6.42394019623897329845170474748, −5.12869402352727076149859832945, −4.25031583491596615901402691977, −2.99499166012987063996440946857, −2.30943307719463265523009320450, −0.889656630307940809558948251958, 0.917310500505801607638230396352, 1.98343660618740599398854094369, 3.55498233033461170668167776205, 4.67907313172413818968165540798, 5.69206158680716832559086112431, 6.24023750829072867883321126113, 7.20671325627383416087269958020, 8.114654301926572707502867249660, 8.699382490472803457602230680728, 9.206508380185648122553916171872

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