Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.896 - 0.443i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 0.885i)2-s + (0.430 + 1.95i)4-s + 2.99i·5-s + 7-s + (−1.25 + 2.53i)8-s + (−2.64 + 3.29i)10-s + 2.13i·11-s + 0.665i·13-s + (1.10 + 0.885i)14-s + (−3.62 + 1.68i)16-s + 7.04·17-s − 1.39i·19-s + (−5.83 + 1.28i)20-s + (−1.89 + 2.35i)22-s + 0.184·23-s + ⋯
L(s)  = 1  + (0.779 + 0.626i)2-s + (0.215 + 0.976i)4-s + 1.33i·5-s + 0.377·7-s + (−0.443 + 0.896i)8-s + (−0.837 + 1.04i)10-s + 0.645i·11-s + 0.184i·13-s + (0.294 + 0.236i)14-s + (−0.907 + 0.420i)16-s + 1.70·17-s − 0.320i·19-s + (−1.30 + 0.287i)20-s + (−0.404 + 0.502i)22-s + 0.0384·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.896 - 0.443i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ -0.896 - 0.443i)$
$L(1)$  $\approx$  $2.623095711$
$L(\frac12)$  $\approx$  $2.623095711$
$L(\frac{3}{2})$   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 \( 1 + (-1.10 - 0.885i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 - 2.99iT - 5T^{2} \)
11 \( 1 - 2.13iT - 11T^{2} \)
13 \( 1 - 0.665iT - 13T^{2} \)
17 \( 1 - 7.04T + 17T^{2} \)
19 \( 1 + 1.39iT - 19T^{2} \)
23 \( 1 - 0.184T + 23T^{2} \)
29 \( 1 - 1.27iT - 29T^{2} \)
31 \( 1 + 7.62T + 31T^{2} \)
37 \( 1 + 6.69iT - 37T^{2} \)
41 \( 1 + 0.274T + 41T^{2} \)
43 \( 1 - 2.63iT - 43T^{2} \)
47 \( 1 + 5.77T + 47T^{2} \)
53 \( 1 + 7.48iT - 53T^{2} \)
59 \( 1 - 9.03iT - 59T^{2} \)
61 \( 1 - 9.80iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 - 7.93T + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 - 7.98T + 79T^{2} \)
83 \( 1 + 6.76iT - 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 0.107T + 97T^{2} \)
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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

−9.869663515472978139521016565633, −8.890828037333211885690385076936, −7.72165464026491167993560267621, −7.39120824251154499102360064224, −6.60993218110159278713290495448, −5.73982690098848021962811611290, −4.94914103599489286577160420263, −3.80177045803990635963566614735, −3.10641217215800797685955605617, −2.02596906695628852353008408433, 0.823915368110465629268879838629, 1.71799493217036957665879808139, 3.18222772723133568763183667197, 3.98649970290695518384638750742, 5.11256984266111985247234482826, 5.39247178831828328231474861521, 6.36165596135735590675354966716, 7.67276325871522199121844590132, 8.389631420360916557120919259107, 9.289029614498255447460692680766

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