Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $0.0460 + 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.725 − 1.21i)2-s + (−0.946 + 1.76i)4-s − 3.06i·5-s − 7-s + (2.82 − 0.130i)8-s + (−3.72 + 2.22i)10-s + 5.80i·11-s − 3.52i·13-s + (0.725 + 1.21i)14-s + (−2.20 − 3.33i)16-s + 6.79·17-s + 5.28i·19-s + (5.40 + 2.90i)20-s + (7.04 − 4.21i)22-s + 5.65·23-s + ⋯
L(s)  = 1  + (−0.513 − 0.858i)2-s + (−0.473 + 0.880i)4-s − 1.37i·5-s − 0.377·7-s + (0.998 − 0.0460i)8-s + (−1.17 + 0.704i)10-s + 1.75i·11-s − 0.977i·13-s + (0.193 + 0.324i)14-s + (−0.552 − 0.833i)16-s + 1.64·17-s + 1.21i·19-s + (1.20 + 0.649i)20-s + (1.50 − 0.898i)22-s + 1.17·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0460 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.0460 + 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ 0.0460 + 0.998i)$
$L(1)$  $\approx$  $1.221885162$
$L(\frac12)$  $\approx$  $1.221885162$
$L(\frac{3}{2})$   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 \( 1 + (0.725 + 1.21i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 3.06iT - 5T^{2} \)
11 \( 1 - 5.80iT - 11T^{2} \)
13 \( 1 + 3.52iT - 13T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 1.21iT - 29T^{2} \)
31 \( 1 - 0.107T + 31T^{2} \)
37 \( 1 + 4.90iT - 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 1.85iT - 43T^{2} \)
47 \( 1 + 6.76T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 7.85iT - 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 + 6.66iT - 67T^{2} \)
71 \( 1 + 1.98T + 71T^{2} \)
73 \( 1 + 6.52T + 73T^{2} \)
79 \( 1 - 2.30T + 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 7.79T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−9.457465752428214036867854227029, −8.655026905193506723522015665821, −7.77951755864335816435317223085, −7.34149240857500282292008950729, −5.72458528181973677387572018934, −4.94760840066310573486384669081, −4.11762360660082688318645492629, −3.09937196271238162781989032964, −1.78574651509908858551018899872, −0.821706155197072658631174586551, 0.954412163671556532903735941540, 2.78065375442855659265928124100, 3.53306916629409168023559421879, 4.91659957233708011093522650581, 5.95617080520689987938095016450, 6.47316865788794658396220357334, 7.19158176085519615876423734171, 7.927459240607901749117863542828, 8.902985626571017270084047646998, 9.490345824338296741959192264001

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