Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.858 - 0.513i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.06i·5-s + 7-s + 5.80i·11-s + 3.52i·13-s + 6.79·17-s + 5.28i·19-s − 5.65·23-s − 4.41·25-s + 1.21i·29-s − 0.107·31-s + 3.06i·35-s + 4.90i·37-s + 11.6·41-s − 1.85i·43-s + 6.76·47-s + ⋯
L(s)  = 1  + 1.37i·5-s + 0.377·7-s + 1.75i·11-s + 0.977i·13-s + 1.64·17-s + 1.21i·19-s − 1.17·23-s − 0.883·25-s + 0.225i·29-s − 0.0192·31-s + 0.518i·35-s + 0.805i·37-s + 1.82·41-s − 0.283i·43-s + 0.986·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.858 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.858 - 0.513i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (3025, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.858 - 0.513i)$
$L(1)$  $\approx$  $2.099727064$
$L(\frac12)$  $\approx$  $2.099727064$
$L(\frac{3}{2})$   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 \( 1 - 3.06iT - 5T^{2} \)
11 \( 1 - 5.80iT - 11T^{2} \)
13 \( 1 - 3.52iT - 13T^{2} \)
17 \( 1 - 6.79T + 17T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 - 1.21iT - 29T^{2} \)
31 \( 1 + 0.107T + 31T^{2} \)
37 \( 1 - 4.90iT - 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 1.85iT - 43T^{2} \)
47 \( 1 - 6.76T + 47T^{2} \)
53 \( 1 - 11.4iT - 53T^{2} \)
59 \( 1 - 7.85iT - 59T^{2} \)
61 \( 1 + 12.0iT - 61T^{2} \)
67 \( 1 + 6.66iT - 67T^{2} \)
71 \( 1 - 1.98T + 71T^{2} \)
73 \( 1 + 6.52T + 73T^{2} \)
79 \( 1 + 2.30T + 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 7.79T + 89T^{2} \)
97 \( 1 - 4.05T + 97T^{2} \)
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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

−8.068256659833425693985898802882, −7.49618454293350801034562874767, −7.16902824639950210094978951725, −6.22452720711417038889495616601, −5.72322504064145012681557026302, −4.57848900766140629009810680896, −4.01023008883300416014376981797, −3.12264427086933459314815240446, −2.18537927280537180198339739725, −1.52058632398388125349210657854, 0.64371242796815150789677727192, 1.00119368716194071473003383782, 2.42524928775423972724596232778, 3.37207156467672864500140103930, 4.12156319598232520449461399118, 5.05107523960616669557144887919, 5.67781624708471757336309236511, 5.95226256236727521622885837450, 7.29259110650929122645287269158, 8.063045673395509952652582751129

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