Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.338 - 0.941i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.338 - 0.941i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.338 - 0.941i)$
$L(1)$  $\approx$  $1.110771296$
$L(\frac12)$  $\approx$  $1.110771296$
$L(\frac{3}{2})$   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

−8.365525639266147217107954006010, −7.928785598988441716919839229351, −7.12174895944114040379507765823, −6.51259304011543504212401882151, −5.40650126742195712975584644114, −4.99406806400983032084371110222, −4.06136364528325738260646654341, −2.89375460811191608295065659019, −2.44975354871639299684087734321, −0.915705536664015264282968140847, 0.38394634284622870023790404919, 1.84588824039898475617730849268, 2.82314802671847226916580070901, 3.56088163808264949608519965661, 4.62269922486504980523440703900, 5.26419297999359944542705753432, 6.10806304366655005373328949834, 6.90954354085569822753901011366, 7.50938592735064141262817723928, 8.271663233198179644292660699396

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