Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.0676 - 0.997i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.12 − 2.12i)5-s − 7-s + (−4.03 − 4.03i)11-s + (−4.91 + 4.91i)13-s + 4.76i·17-s + (2.21 + 2.21i)19-s − 6.91i·23-s − 3.99i·25-s + (2.61 + 2.61i)29-s + 0.712i·31-s + (−2.12 + 2.12i)35-s + (5.73 + 5.73i)37-s − 3.59·41-s + (−3.36 + 3.36i)43-s − 6.04·47-s + ⋯
L(s)  = 1  + (0.948 − 0.948i)5-s − 0.377·7-s + (−1.21 − 1.21i)11-s + (−1.36 + 1.36i)13-s + 1.15i·17-s + (0.508 + 0.508i)19-s − 1.44i·23-s − 0.798i·25-s + (0.485 + 0.485i)29-s + 0.127i·31-s + (−0.358 + 0.358i)35-s + (0.943 + 0.943i)37-s − 0.561·41-s + (−0.512 + 0.512i)43-s − 0.881·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0676 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0676 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0676 - 0.997i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (3599, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.0676 - 0.997i)$
$L(1)$  $\approx$  $0.9685611216$
$L(\frac12)$  $\approx$  $0.9685611216$
$L(\frac{3}{2})$   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 \( 1 + (-2.12 + 2.12i)T - 5iT^{2} \)
11 \( 1 + (4.03 + 4.03i)T + 11iT^{2} \)
13 \( 1 + (4.91 - 4.91i)T - 13iT^{2} \)
17 \( 1 - 4.76iT - 17T^{2} \)
19 \( 1 + (-2.21 - 2.21i)T + 19iT^{2} \)
23 \( 1 + 6.91iT - 23T^{2} \)
29 \( 1 + (-2.61 - 2.61i)T + 29iT^{2} \)
31 \( 1 - 0.712iT - 31T^{2} \)
37 \( 1 + (-5.73 - 5.73i)T + 37iT^{2} \)
41 \( 1 + 3.59T + 41T^{2} \)
43 \( 1 + (3.36 - 3.36i)T - 43iT^{2} \)
47 \( 1 + 6.04T + 47T^{2} \)
53 \( 1 + (-4.53 + 4.53i)T - 53iT^{2} \)
59 \( 1 + (-4.82 - 4.82i)T + 59iT^{2} \)
61 \( 1 + (6.72 - 6.72i)T - 61iT^{2} \)
67 \( 1 + (-3.87 - 3.87i)T + 67iT^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 + 4.61iT - 73T^{2} \)
79 \( 1 - 7.43iT - 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 - 4.23T + 89T^{2} \)
97 \( 1 + 6.76T + 97T^{2} \)
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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

−8.437084638965716631028659459114, −8.270222856135999742480001696935, −7.02857368061079421164349448082, −6.32774493414585823601206556811, −5.59877269170262653435527894083, −4.97391846367961580295645770288, −4.22823884087753809456977157196, −2.98651191626828828788283456645, −2.21710526512873406522080852505, −1.15324095048480031092606251998, 0.27202978332124667606291532736, 2.07986900576505560633153911818, 2.68370495734650617755153155693, 3.25527261118402393275991776978, 4.82404275922376935126954932155, 5.22793319644013248814239826917, 5.98629744140367725147105921400, 7.02290014065078743905923414642, 7.40628825841240342560122091437, 7.962178671281606079068147748525

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