Properties

Degree 2
Conductor $ 2^{5} \cdot 3^{3} \cdot 7 $
Sign $-0.482 + 0.875i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.381·5-s + i·7-s + 2.13i·11-s + 6.49i·13-s − 6.97i·17-s − 6.19·19-s + 1.82·23-s − 4.85·25-s + 0.694·29-s − 1.67i·31-s + 0.381i·35-s − 8.06i·37-s + 1.31i·41-s − 7.67·43-s − 6.82·47-s + ⋯
L(s)  = 1  + 0.170·5-s + 0.377i·7-s + 0.643i·11-s + 1.80i·13-s − 1.69i·17-s − 1.42·19-s + 0.380·23-s − 0.970·25-s + 0.129·29-s − 0.301i·31-s + 0.0644i·35-s − 1.32i·37-s + 0.204i·41-s − 1.17·43-s − 0.995·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6048\)    =    \(2^{5} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.482 + 0.875i$
motivic weight  =  \(1\)
character  :  $\chi_{6048} (5615, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 6048,\ (\ :1/2),\ -0.482 + 0.875i)$
$L(1)$  $\approx$  $0.5213128330$
$L(\frac12)$  $\approx$  $0.5213128330$
$L(\frac{3}{2})$   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 \( 1 - 0.381T + 5T^{2} \)
11 \( 1 - 2.13iT - 11T^{2} \)
13 \( 1 - 6.49iT - 13T^{2} \)
17 \( 1 + 6.97iT - 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 - 1.82T + 23T^{2} \)
29 \( 1 - 0.694T + 29T^{2} \)
31 \( 1 + 1.67iT - 31T^{2} \)
37 \( 1 + 8.06iT - 37T^{2} \)
41 \( 1 - 1.31iT - 41T^{2} \)
43 \( 1 + 7.67T + 43T^{2} \)
47 \( 1 + 6.82T + 47T^{2} \)
53 \( 1 + 0.954T + 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 + 12.4iT - 61T^{2} \)
67 \( 1 + 0.634T + 67T^{2} \)
71 \( 1 + 5.79T + 71T^{2} \)
73 \( 1 - 8.13T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + 3.36iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−7.74911273861768013167613850952, −7.04223674713090738592598499251, −6.53432687492906609463583320476, −5.76570152367776408252251251782, −4.71746070402366046709730000552, −4.44130776527571590509558143265, −3.35194050381337937094250117482, −2.23805386499072799038433281840, −1.81154292538306704923872175775, −0.13264207384251671159826389810, 1.10618231408343695992888506152, 2.13636408269797885033680297265, 3.23122573554382312485318815340, 3.75059217019639796382523661284, 4.73311339060037879291431368018, 5.54216212979595990196522182303, 6.19919322601900481133559858933, 6.72070143145098319105329387874, 7.893179045362077662331265013535, 8.219332860774598142220758605323

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