Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} \cdot 7 $
Sign $-0.938 - 0.345i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.671 + 1.59i)3-s + (−0.158 + 0.900i)5-s + (−0.241 − 2.63i)7-s + (−2.09 + 2.14i)9-s + (−3.68 + 0.650i)11-s + (−2.48 + 2.96i)13-s + (−1.54 + 0.351i)15-s − 6.10·17-s + 5.56i·19-s + (4.04 − 2.15i)21-s + (−1.03 + 1.22i)23-s + (3.91 + 1.42i)25-s + (−4.83 − 1.90i)27-s + (4.21 + 5.02i)29-s + (−0.823 − 2.26i)31-s + ⋯
L(s)  = 1  + (0.387 + 0.921i)3-s + (−0.0710 + 0.402i)5-s + (−0.0913 − 0.995i)7-s + (−0.699 + 0.714i)9-s + (−1.11 + 0.196i)11-s + (−0.688 + 0.820i)13-s + (−0.398 + 0.0907i)15-s − 1.48·17-s + 1.27i·19-s + (0.882 − 0.470i)21-s + (−0.215 + 0.256i)23-s + (0.782 + 0.284i)25-s + (−0.930 − 0.367i)27-s + (0.783 + 0.933i)29-s + (−0.147 − 0.406i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.938 - 0.345i$
motivic weight  =  \(1\)
character  :  $\chi_{756} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 756,\ (\ :1/2),\ -0.938 - 0.345i)\)
\(L(1)\)  \(\approx\)  \(0.154326 + 0.864761i\)
\(L(\frac12)\)  \(\approx\)  \(0.154326 + 0.864761i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.671 - 1.59i)T \)
7 \( 1 + (0.241 + 2.63i)T \)
good5 \( 1 + (0.158 - 0.900i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (3.68 - 0.650i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.48 - 2.96i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 6.10T + 17T^{2} \)
19 \( 1 - 5.56iT - 19T^{2} \)
23 \( 1 + (1.03 - 1.22i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-4.21 - 5.02i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.823 + 2.26i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-1.21 - 2.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.95 + 4.15i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-7.68 - 2.79i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-4.31 - 1.57i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (9.92 - 5.72i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.682 - 0.572i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-4.21 + 11.5i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.26 + 12.8i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-11.1 - 6.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.20 + 2.42i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.979 + 5.55i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (6.26 - 5.25i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 4.08T + 89T^{2} \)
97 \( 1 + (4.13 - 11.3i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.71731827120856739694454970205, −9.943160213960727963158056385315, −9.189535504797101006818576388547, −8.135709564524787326459701322579, −7.37480648220095324008442125131, −6.42453472425576318337787570621, −5.04105803008633708343037075160, −4.35294727255541262638981609275, −3.34093325705219542064707406424, −2.19333656319721451150908877280, 0.39071725367595519650825708344, 2.37049112345911478617215540025, 2.81526832314190451086871053051, 4.65652379254487324880141324544, 5.53369767874669467699493993730, 6.52545567699007263024908093802, 7.41314832758951188859447359028, 8.454907261991683018376810389320, 8.734994595254870019673149053695, 9.823981268413098525876720397378

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