Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.722 - 0.691i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.11·5-s + (1.37 − 2.26i)7-s + 0.812·11-s + 2·13-s + 3.55i·17-s + 4.52i·19-s + 3.63i·23-s − 3.75·25-s + 8.95i·29-s + 5.08·31-s + (1.53 − 2.52i)35-s + 0.718i·37-s + 10.4i·41-s − 1.11·43-s − 8.55·47-s + ⋯
L(s)  = 1  + 0.498·5-s + (0.518 − 0.854i)7-s + 0.245·11-s + 0.554·13-s + 0.862i·17-s + 1.03i·19-s + 0.758i·23-s − 0.751·25-s + 1.66i·29-s + 0.914·31-s + (0.258 − 0.426i)35-s + 0.118i·37-s + 1.63i·41-s − 0.170·43-s − 1.24·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.722 - 0.691i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.722 - 0.691i)$
$L(1)$  $\approx$  $2.192779609$
$L(\frac12)$  $\approx$  $2.192779609$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1.37 + 2.26i)T \)
good5 \( 1 - 1.11T + 5T^{2} \)
11 \( 1 - 0.812T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 3.55iT - 17T^{2} \)
19 \( 1 - 4.52iT - 19T^{2} \)
23 \( 1 - 3.63iT - 23T^{2} \)
29 \( 1 - 8.95iT - 29T^{2} \)
31 \( 1 - 5.08T + 31T^{2} \)
37 \( 1 - 0.718iT - 37T^{2} \)
41 \( 1 - 10.4iT - 41T^{2} \)
43 \( 1 + 1.11T + 43T^{2} \)
47 \( 1 + 8.55T + 47T^{2} \)
53 \( 1 - 5.08iT - 53T^{2} \)
59 \( 1 + 2.23iT - 59T^{2} \)
61 \( 1 - 5.27T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 5.87iT - 71T^{2} \)
73 \( 1 - 3.86iT - 73T^{2} \)
79 \( 1 + 1.70iT - 79T^{2} \)
83 \( 1 + 7.27iT - 83T^{2} \)
89 \( 1 + 3.37iT - 89T^{2} \)
97 \( 1 + 8.55iT - 97T^{2} \)
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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

−8.284866758941006331657743940116, −8.012692284569818276896321134092, −7.00627248253799946341165843911, −6.33276978066671979588501848915, −5.61973879954459401447360442021, −4.75595488412385205166864138548, −3.89609003972498988277714909369, −3.24298680122825870865664901780, −1.78420114939453721766487060887, −1.24307902619873227175361245883, 0.66263600220072627153265327579, 2.05502858326835760358167445805, 2.57867399557314540475117732927, 3.75893271330754917427373112301, 4.73139542284908317273848814685, 5.34285607540264169590813804472, 6.18044338285903302872322598294, 6.71857669174164739440517900704, 7.75283410252268018751986743211, 8.407021964887668924104697709161

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