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  =  \((2,\ 1512,\ (\ :1/2),\ -0.622 + 0.782i)\)
\(L(1)\)  \(\approx\)  \(1.069472355\)
\(L(\frac12)\)  \(\approx\)  \(1.069472355\)
\(L(\frac{3}{2})\)   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

−9.720859777202401755703771678527, −9.090875014038096016566975999632, −8.288978849657160407286781425395, −7.23802393471703311195574951903, −6.70538445630171217391600096026, −6.34079508618146300648815699816, −4.92810242843991471588167394735, −4.31630792827219082440176395334, −3.22519883013167343594031256933, −2.13289239192001868765934972694, 0.37913651592324574856828351379, 1.53410993123720857250350194879, 2.63925566374152648972641735550, 3.99539519069217676755449087522, 4.51067721835657405783062644608, 5.58314830823384045864899681246, 5.95817015721820349518740464294, 7.78421056544743172327229548104, 8.346197987698188051740754466520, 8.937361942699386603960660714375

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