Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.986 - 0.163i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.667 + 0.667i)5-s − 7-s + (−1.57 + 1.57i)11-s + (−1.83 − 1.83i)13-s − 3.40i·17-s + (−3.18 + 3.18i)19-s − 0.793i·23-s − 4.10i·25-s + (−1.73 + 1.73i)29-s + 3.28i·31-s + (−0.667 − 0.667i)35-s + (7.72 − 7.72i)37-s + 7.19·41-s + (5.84 + 5.84i)43-s + 13.0·47-s + ⋯
L(s)  = 1  + (0.298 + 0.298i)5-s − 0.377·7-s + (−0.475 + 0.475i)11-s + (−0.507 − 0.507i)13-s − 0.826i·17-s + (−0.730 + 0.730i)19-s − 0.165i·23-s − 0.821i·25-s + (−0.322 + 0.322i)29-s + 0.589i·31-s + (−0.112 − 0.112i)35-s + (1.26 − 1.26i)37-s + 1.12·41-s + (0.890 + 0.890i)43-s + 1.89·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.986 - 0.163i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1583, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.986 - 0.163i)$
$L(1)$  $\approx$  $1.630735564$
$L(\frac12)$  $\approx$  $1.630735564$
$L(\frac{3}{2})$   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 \( 1 + (-0.667 - 0.667i)T + 5iT^{2} \)
11 \( 1 + (1.57 - 1.57i)T - 11iT^{2} \)
13 \( 1 + (1.83 + 1.83i)T + 13iT^{2} \)
17 \( 1 + 3.40iT - 17T^{2} \)
19 \( 1 + (3.18 - 3.18i)T - 19iT^{2} \)
23 \( 1 + 0.793iT - 23T^{2} \)
29 \( 1 + (1.73 - 1.73i)T - 29iT^{2} \)
31 \( 1 - 3.28iT - 31T^{2} \)
37 \( 1 + (-7.72 + 7.72i)T - 37iT^{2} \)
41 \( 1 - 7.19T + 41T^{2} \)
43 \( 1 + (-5.84 - 5.84i)T + 43iT^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 + (-3.34 - 3.34i)T + 53iT^{2} \)
59 \( 1 + (-7.41 + 7.41i)T - 59iT^{2} \)
61 \( 1 + (-1.93 - 1.93i)T + 61iT^{2} \)
67 \( 1 + (6.38 - 6.38i)T - 67iT^{2} \)
71 \( 1 + 3.41iT - 71T^{2} \)
73 \( 1 - 8.13iT - 73T^{2} \)
79 \( 1 + 0.0502iT - 79T^{2} \)
83 \( 1 + (2.29 + 2.29i)T + 83iT^{2} \)
89 \( 1 + 7.18T + 89T^{2} \)
97 \( 1 - 1.49T + 97T^{2} \)
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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

−8.462366755680587101001902482062, −7.53670582470579655039068906695, −7.18322809834898688784195065183, −6.12122592771007671332154163325, −5.65518346472140478842006799945, −4.65313023264536711559879685343, −3.92892658926707958857551399042, −2.71217255689420248281796053724, −2.30459776869866183447960380434, −0.72165983552045763719296352279, 0.70405330975283684083003366285, 2.04792602654565211293671918281, 2.80180917042682796466960023579, 3.93575374991458261217165381053, 4.57348489082593525301850354603, 5.63756948403780843398071012404, 6.04406688624507550775551119160, 7.01317879659464711579417302309, 7.65131064473700392037327401796, 8.528838717656843752348796720236

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