Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.999 - 0.00496i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0893 + 0.0893i)5-s − 7-s + (2.42 − 2.42i)11-s + (−3.86 − 3.86i)13-s − 0.794i·17-s + (2.65 − 2.65i)19-s − 3.92i·23-s − 4.98i·25-s + (−7.47 + 7.47i)29-s + 5.55i·31-s + (−0.0893 − 0.0893i)35-s + (−6.35 + 6.35i)37-s − 6.96·41-s + (1.25 + 1.25i)43-s + 6.48·47-s + ⋯
L(s)  = 1  + (0.0399 + 0.0399i)5-s − 0.377·7-s + (0.731 − 0.731i)11-s + (−1.07 − 1.07i)13-s − 0.192i·17-s + (0.608 − 0.608i)19-s − 0.817i·23-s − 0.996i·25-s + (−1.38 + 1.38i)29-s + 0.997i·31-s + (−0.0151 − 0.0151i)35-s + (−1.04 + 1.04i)37-s − 1.08·41-s + (0.191 + 0.191i)43-s + 0.945·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.999 - 0.00496i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.999 - 0.00496i)$
$L(1)$  $\approx$  $0.3587829950$
$L(\frac12)$  $\approx$  $0.3587829950$
$L(\frac{3}{2})$   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 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + (-0.0893 - 0.0893i)T + 5iT^{2} \)
11 \( 1 + (-2.42 + 2.42i)T - 11iT^{2} \)
13 \( 1 + (3.86 + 3.86i)T + 13iT^{2} \)
17 \( 1 + 0.794iT - 17T^{2} \)
19 \( 1 + (-2.65 + 2.65i)T - 19iT^{2} \)
23 \( 1 + 3.92iT - 23T^{2} \)
29 \( 1 + (7.47 - 7.47i)T - 29iT^{2} \)
31 \( 1 - 5.55iT - 31T^{2} \)
37 \( 1 + (6.35 - 6.35i)T - 37iT^{2} \)
41 \( 1 + 6.96T + 41T^{2} \)
43 \( 1 + (-1.25 - 1.25i)T + 43iT^{2} \)
47 \( 1 - 6.48T + 47T^{2} \)
53 \( 1 + (-0.620 - 0.620i)T + 53iT^{2} \)
59 \( 1 + (3.39 - 3.39i)T - 59iT^{2} \)
61 \( 1 + (-7.51 - 7.51i)T + 61iT^{2} \)
67 \( 1 + (2.22 - 2.22i)T - 67iT^{2} \)
71 \( 1 + 7.95iT - 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + (11.2 + 11.2i)T + 83iT^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 4.33T + 97T^{2} \)
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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

−8.228410154510407681720002099476, −7.00211720934420719493941539040, −6.94541334594460040103312927981, −5.70945973231814285174722695312, −5.22414299407557365048236030421, −4.25753846902073955341714670152, −3.21832689375428929590443473424, −2.72236725968002332433151125407, −1.30879871505852515714576566788, −0.10108859039651993587865423513, 1.61056982477162599350394124975, 2.27851801938706190456884049902, 3.64918983280735603209408246564, 4.07784826325104794929385981288, 5.15305269686634271993062660658, 5.79920110579522293786660339291, 6.76574856340606910587593863154, 7.30703774716235706119488042949, 7.88563265340242269604506904505, 9.108087121641443717853145341253

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