Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.701 - 0.712i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3 − 5.19i)11-s − 2·13-s + (−2 + 3.46i)19-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 6·29-s + (4 + 6.92i)31-s + (−1 + 1.73i)37-s + 12·41-s − 4·43-s + (−6 + 10.3i)47-s + (−3 − 5.19i)53-s + (−5 + 8.66i)61-s + (−4 − 6.92i)67-s − 6·71-s + ⋯
L(s)  = 1  + (−0.904 − 1.56i)11-s − 0.554·13-s + (−0.458 + 0.794i)19-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 1.11·29-s + (0.718 + 1.24i)31-s + (−0.164 + 0.284i)37-s + 1.87·41-s − 0.609·43-s + (−0.875 + 1.51i)47-s + (−0.412 − 0.713i)53-s + (−0.640 + 1.10i)61-s + (−0.488 − 0.846i)67-s − 0.712·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.701 - 0.712i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1549, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.701 - 0.712i)\)
\(L(1)\)  \(\approx\)  \(0.4779663785\)
\(L(\frac12)\)  \(\approx\)  \(0.4779663785\)
\(L(\frac{3}{2})\)   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 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−9.554574195003549911579618090288, −8.773650660616631093854656284318, −7.960665559929633233705723364521, −7.42644860065159425037676445151, −6.20080554217265571732577823089, −5.64655303408395581646025433378, −4.76355785055075911897485270082, −3.55129892550125268171494229195, −2.85389113381592196300907702039, −1.46082212216262558740401482019, 0.17057248429115436909393366527, 2.07900445752771484411713967524, 2.68751287498481909050364269498, 4.27818700991027018142613131213, 4.69279063319515618283618631482, 5.74946873491688202489108732054, 6.72575848984387405197821086068, 7.43459518678917806391853825802, 8.115740235985354181582674125923, 9.057669002120242481862504988676

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