Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $0.707 - 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.14i·5-s i·7-s + 1.41·11-s + 0.449·13-s + 2.51i·17-s + 2.44i·19-s − 5.65·23-s − 4.89·25-s + 8.34i·29-s + 1.55i·31-s − 3.14·35-s + 9·37-s + 10.0i·41-s + 8.34i·43-s − 0.460·47-s + ⋯
L(s)  = 1  − 1.40i·5-s − 0.377i·7-s + 0.426·11-s + 0.124·13-s + 0.608i·17-s + 0.561i·19-s − 1.17·23-s − 0.979·25-s + 1.54i·29-s + 0.278i·31-s − 0.531·35-s + 1.47·37-s + 1.57i·41-s + 1.27i·43-s − 0.0672·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ 0.707 - 0.707i)$
$L(1)$  $\approx$  $1.417032918$
$L(\frac12)$  $\approx$  $1.417032918$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 3.14iT - 5T^{2} \)
11 \( 1 - 1.41T + 11T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 - 2.51iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 8.34iT - 29T^{2} \)
31 \( 1 - 1.55iT - 31T^{2} \)
37 \( 1 - 9T + 37T^{2} \)
41 \( 1 - 10.0iT - 41T^{2} \)
43 \( 1 - 8.34iT - 43T^{2} \)
47 \( 1 + 0.460T + 47T^{2} \)
53 \( 1 - 5.02iT - 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 5.55T + 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 9.79T + 73T^{2} \)
79 \( 1 - 0.348iT - 79T^{2} \)
83 \( 1 + 2.36T + 83T^{2} \)
89 \( 1 + 1.55iT - 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−8.022574431226840044227116539195, −7.81805603867569098139805871904, −6.59233943723062745994363894773, −6.04742797466770544209926880633, −5.22740169917542588800975254649, −4.46174661187780724965442124356, −3.99065311733299973274966688552, −2.96356768244291460123816524908, −1.59713539203354763884373306138, −1.10649070392080508803323222944, 0.38409991216973001025554956948, 2.05211161776278149583071426150, 2.57860132009858808037705670405, 3.50648176915530664283888849987, 4.17428664376833229601595568041, 5.18251293407405123340489082103, 6.18871567353129440366402976078, 6.37269793498989187416297795351, 7.35241282308613114854482692925, 7.76906569798388112834364854546

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