Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.686 - 0.726i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.925 − 0.925i)5-s − 7-s + (1.72 + 1.72i)11-s + (0.328 − 0.328i)13-s − 2.34i·17-s + (1.77 + 1.77i)19-s + 6.17i·23-s + 3.28i·25-s + (−0.122 − 0.122i)29-s + 1.74i·31-s + (−0.925 + 0.925i)35-s + (−1.68 − 1.68i)37-s + 2.88·41-s + (−2.77 + 2.77i)43-s − 5.92·47-s + ⋯
L(s)  = 1  + (0.413 − 0.413i)5-s − 0.377·7-s + (0.519 + 0.519i)11-s + (0.0910 − 0.0910i)13-s − 0.567i·17-s + (0.408 + 0.408i)19-s + 1.28i·23-s + 0.657i·25-s + (−0.0227 − 0.0227i)29-s + 0.313i·31-s + (−0.156 + 0.156i)35-s + (−0.276 − 0.276i)37-s + 0.451·41-s + (−0.422 + 0.422i)43-s − 0.863·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.686 - 0.726i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.686 - 0.726i)$
$L(1)$  $\approx$  $1.842751817$
$L(\frac12)$  $\approx$  $1.842751817$
$L(\frac{3}{2})$   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 \( 1 + (-0.925 + 0.925i)T - 5iT^{2} \)
11 \( 1 + (-1.72 - 1.72i)T + 11iT^{2} \)
13 \( 1 + (-0.328 + 0.328i)T - 13iT^{2} \)
17 \( 1 + 2.34iT - 17T^{2} \)
19 \( 1 + (-1.77 - 1.77i)T + 19iT^{2} \)
23 \( 1 - 6.17iT - 23T^{2} \)
29 \( 1 + (0.122 + 0.122i)T + 29iT^{2} \)
31 \( 1 - 1.74iT - 31T^{2} \)
37 \( 1 + (1.68 + 1.68i)T + 37iT^{2} \)
41 \( 1 - 2.88T + 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 + 5.92T + 47T^{2} \)
53 \( 1 + (0.973 - 0.973i)T - 53iT^{2} \)
59 \( 1 + (-8.33 - 8.33i)T + 59iT^{2} \)
61 \( 1 + (-4.28 + 4.28i)T - 61iT^{2} \)
67 \( 1 + (1.78 + 1.78i)T + 67iT^{2} \)
71 \( 1 - 8.57iT - 71T^{2} \)
73 \( 1 + 6.41iT - 73T^{2} \)
79 \( 1 - 5.38iT - 79T^{2} \)
83 \( 1 + (-3.46 + 3.46i)T - 83iT^{2} \)
89 \( 1 + 1.51T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.670592425554653360285862814509, −7.69544899595259949232672514560, −7.13330361240974570822453279090, −6.30815139057644390777339663108, −5.50805785516027622256327176410, −4.92376638374665101377319056046, −3.87942763385023385127896116137, −3.15984690175265060427268687285, −1.97694876345710371465655397954, −1.08328010169342176431408113220, 0.59147531826747911614526904926, 1.92263327066137276377564439305, 2.84001898867382272307794087917, 3.66118170746216019343678256306, 4.52614188731596816026249406666, 5.47382208522676535975885855792, 6.39586366931403908321719035597, 6.57702868561140702740533945017, 7.59559154866287931920520247117, 8.501341569141830484766133091565

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