Properties

Degree 2
Conductor $ 2^{3} \cdot 3^{3} \cdot 7 $
Sign $-0.622 + 0.782i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.417 − 1.35i)2-s + (−1.65 + 1.12i)4-s − 3.04i·5-s + 7-s + (2.21 + 1.76i)8-s + (−4.11 + 1.27i)10-s − 0.128i·11-s + 6.30i·13-s + (−0.417 − 1.35i)14-s + (1.45 − 3.72i)16-s + 5.32·17-s − 6.68i·19-s + (3.43 + 5.02i)20-s + (−0.173 + 0.0536i)22-s + 5.18·23-s + ⋯
L(s)  = 1  + (−0.295 − 0.955i)2-s + (−0.825 + 0.563i)4-s − 1.36i·5-s + 0.377·7-s + (0.782 + 0.622i)8-s + (−1.30 + 0.401i)10-s − 0.0387i·11-s + 1.74i·13-s + (−0.111 − 0.361i)14-s + (0.364 − 0.931i)16-s + 1.29·17-s − 1.53i·19-s + (0.767 + 1.12i)20-s + (−0.0370 + 0.0114i)22-s + 1.08·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.622 + 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.622 + 0.782i$
motivic weight  =  \(1\)
character  :  $\chi_{1512} (757, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1512,\ (\ :1/2),\ -0.622 + 0.782i)$
$L(1)$  $\approx$  $1.403584709$
$L(\frac12)$  $\approx$  $1.403584709$
$L(\frac{3}{2})$   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 \( 1 + (0.417 + 1.35i)T \)
3 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + 3.04iT - 5T^{2} \)
11 \( 1 + 0.128iT - 11T^{2} \)
13 \( 1 - 6.30iT - 13T^{2} \)
17 \( 1 - 5.32T + 17T^{2} \)
19 \( 1 + 6.68iT - 19T^{2} \)
23 \( 1 - 5.18T + 23T^{2} \)
29 \( 1 + 9.96iT - 29T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 + 0.796iT - 37T^{2} \)
41 \( 1 + 2.96T + 41T^{2} \)
43 \( 1 - 6.99iT - 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 - 1.14iT - 53T^{2} \)
59 \( 1 - 11.0iT - 59T^{2} \)
61 \( 1 + 14.6iT - 61T^{2} \)
67 \( 1 + 1.05iT - 67T^{2} \)
71 \( 1 - 1.10T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 - 2.62T + 79T^{2} \)
83 \( 1 + 6.46iT - 83T^{2} \)
89 \( 1 + 2.23T + 89T^{2} \)
97 \( 1 + 7.88T + 97T^{2} \)
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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

−9.306320522176950767989080207497, −8.637796213120125758239882310560, −7.925833803514153598364851603198, −6.94536355237012220880542419704, −5.58023073518695255840194222306, −4.59867170515081959734423772547, −4.32956577858982911451614428110, −2.89031640512347829407974589498, −1.70188265046558843351160076544, −0.75567477727282602723483056131, 1.21727783836263282151366398156, 2.98394747099775275094828451626, 3.70205933726832517382719254855, 5.23471775607649576176732799944, 5.65041237503617825304180122205, 6.66678814398328910405422611376, 7.40384039062370695152546251162, 7.945281466981359765164996162454, 8.713797156394441408482501797067, 9.908719906462655435508411524385

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