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  =  \((2,\ 1512,\ (\ :1/2),\ 0.935 + 0.352i)\)
\(L(1)\)  \(\approx\)  \(2.546447418\)
\(L(\frac12)\)  \(\approx\)  \(2.546447418\)
\(L(\frac{3}{2})\)   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

−8.956471110631472011734586193042, −8.527132538366501898555745502876, −7.87486725002920841370515109746, −6.87057501736891212267111295694, −6.02609648764282503601156482038, −5.24212289697588582876000739974, −4.67497543761630835555633636372, −3.31120813428015086999828418343, −2.80031094928492974397997673533, −0.77621812551885323075686197775, 1.69017686352283918033503508013, 2.24351581010220567615907674350, 3.63524681279116293016606993278, 4.51319745496531945408766962226, 4.98210427922203630794164721046, 6.42922251426175282034163872223, 6.76963621312381506815844490762, 7.72326968875456927743510121306, 9.040461082685529859232102987044, 9.668767498746022244798752878480

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