Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $-0.639 - 0.769i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.09·5-s i·7-s − 0.646i·11-s + 4.37i·13-s − 0.295i·17-s + 5.29·19-s − 3.94·23-s + 4.58·25-s + 5.03·29-s − 9.16i·31-s + 3.09i·35-s + 2.55i·37-s − 10.4i·41-s + 1.63·43-s − 5.65·47-s + ⋯
L(s)  = 1  − 1.38·5-s − 0.377i·7-s − 0.194i·11-s + 1.21i·13-s − 0.0715i·17-s + 1.21·19-s − 0.823·23-s + 0.916·25-s + 0.934·29-s − 1.64i·31-s + 0.523i·35-s + 0.419i·37-s − 1.62i·41-s + 0.249·43-s − 0.825·47-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.639 - 0.769i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ -0.639 - 0.769i)$
$L(1)$  $\approx$  $0.4630843679$
$L(\frac12)$  $\approx$  $0.4630843679$
$L(\frac{3}{2})$   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 \( 1 + iT \)
good5 \( 1 + 3.09T + 5T^{2} \)
11 \( 1 + 0.646iT - 11T^{2} \)
13 \( 1 - 4.37iT - 13T^{2} \)
17 \( 1 + 0.295iT - 17T^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
23 \( 1 + 3.94T + 23T^{2} \)
29 \( 1 - 5.03T + 29T^{2} \)
31 \( 1 + 9.16iT - 31T^{2} \)
37 \( 1 - 2.55iT - 37T^{2} \)
41 \( 1 + 10.4iT - 41T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 8.64T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 - 2.55T + 67T^{2} \)
71 \( 1 - 1.11T + 71T^{2} \)
73 \( 1 + 9.58T + 73T^{2} \)
79 \( 1 - 16.7iT - 79T^{2} \)
83 \( 1 - 8.50iT - 83T^{2} \)
89 \( 1 - 10.4iT - 89T^{2} \)
97 \( 1 + 2.41T + 97T^{2} \)
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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

−8.515891503369569878324573355038, −7.943453840062460838014223094976, −7.31395821233384324125278896188, −6.70392570038257813670615959632, −5.76672785841780206376177697387, −4.73699098254377015498527578635, −4.06817433445988265856862522977, −3.53541666551531831774256477319, −2.40804446558771210072405662354, −1.06934599763782384526145005732, 0.15906754049462487837373373401, 1.42702576049018909315258581933, 3.00833403766688459668699304043, 3.30984088760659586656983702990, 4.44136101582430613400638149554, 5.05485832815298037933186914159, 5.94745640703027056928136861210, 6.82773391382860713193043230845, 7.68588582078155981718576607284, 8.032699405381475211855996619825

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