Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{3} \cdot 7 $
Sign $0.116 - 0.993i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 + 0.373i)3-s + (−0.203 + 1.15i)5-s + (2.02 + 1.70i)7-s + (2.72 − 1.26i)9-s + (3.15 − 0.555i)11-s + (−1.12 + 1.33i)13-s + (−0.0867 − 2.02i)15-s − 3.16·17-s − 6.65i·19-s + (−4.05 − 2.12i)21-s + (−3.65 + 4.35i)23-s + (3.40 + 1.23i)25-s + (−4.13 + 3.15i)27-s + (3.84 + 4.58i)29-s + (1.52 + 4.18i)31-s + ⋯
L(s)  = 1  + (−0.976 + 0.215i)3-s + (−0.0910 + 0.516i)5-s + (0.764 + 0.644i)7-s + (0.907 − 0.420i)9-s + (0.950 − 0.167i)11-s + (−0.311 + 0.371i)13-s + (−0.0223 − 0.524i)15-s − 0.767·17-s − 1.52i·19-s + (−0.885 − 0.464i)21-s + (−0.762 + 0.908i)23-s + (0.681 + 0.247i)25-s + (−0.795 + 0.606i)27-s + (0.714 + 0.851i)29-s + (0.273 + 0.751i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.116 - 0.993i$
motivic weight  =  \(1\)
character  :  $\chi_{756} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 756,\ (\ :1/2),\ 0.116 - 0.993i)\)
\(L(1)\)  \(\approx\)  \(0.839220 + 0.746237i\)
\(L(\frac12)\)  \(\approx\)  \(0.839220 + 0.746237i\)
\(L(\frac{3}{2})\)   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 \( 1 + (1.69 - 0.373i)T \)
7 \( 1 + (-2.02 - 1.70i)T \)
good5 \( 1 + (0.203 - 1.15i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (-3.15 + 0.555i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (1.12 - 1.33i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + 3.16T + 17T^{2} \)
19 \( 1 + 6.65iT - 19T^{2} \)
23 \( 1 + (3.65 - 4.35i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.84 - 4.58i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.52 - 4.18i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (-4.75 - 8.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.74 - 6.50i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-3.80 - 1.38i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.95 + 2.89i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (11.7 - 6.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.39 + 7.04i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (0.888 - 2.43i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.66 + 15.1i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.08 - 1.20i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.01 - 0.583i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.74 - 9.88i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (5.29 - 4.44i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 + (0.0688 - 0.189i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.00409765981239111366791649104, −9.614210705026546637518906384462, −9.068904127206953489168608155163, −7.88540454372170516517321959469, −6.73154449566900846623486800792, −6.33164807789700937872817115626, −5.00060796657120799878710786494, −4.47655502565090426828251524189, −2.98212358129779391864313372423, −1.42851273427336354772158242967, 0.72545918001127836238797095549, 1.97975906015787284317736904516, 4.15681747109737653095107456069, 4.51975108145811861831000324368, 5.77638452826596442753689338059, 6.50968854197363188147306657949, 7.59173802161385141636601768094, 8.223455772682795802609823181488, 9.425847798241634705755079327116, 10.33302539583987161953661035146

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