Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $0.918 + 0.394i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.686 − 1.18i)5-s + (−0.5 + 0.866i)7-s + 0.999·8-s + 1.37·10-s + (2.18 − 3.78i)11-s + (−1 − 1.73i)13-s + (−0.499 − 0.866i)14-s + (−0.5 + 0.866i)16-s + 4.37·17-s + 5·19-s + (−0.686 + 1.18i)20-s + (2.18 + 3.78i)22-s + (−3.68 − 6.38i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.306 − 0.531i)5-s + (−0.188 + 0.327i)7-s + 0.353·8-s + 0.433·10-s + (0.659 − 1.14i)11-s + (−0.277 − 0.480i)13-s + (−0.133 − 0.231i)14-s + (−0.125 + 0.216i)16-s + 1.06·17-s + 1.14·19-s + (−0.153 + 0.265i)20-s + (0.466 + 0.807i)22-s + (−0.768 − 1.33i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.918 + 0.394i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (127, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ 0.918 + 0.394i)$
$L(1)$  $\approx$  $0.996713 - 0.204789i$
$L(\frac12)$  $\approx$  $0.996713 - 0.204789i$
$L(\frac{3}{2})$   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 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.686 + 1.18i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.18 + 3.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (3.68 + 6.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.37 + 2.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-5.18 - 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.74T + 53T^{2} \)
59 \( 1 + (-3.55 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.05 + 12.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.55 + 13.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 + (6.05 - 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.74 - 4.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.25T + 89T^{2} \)
97 \( 1 + (4.55 - 7.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.36455164675024265570475509744, −10.13793023661645728921978327955, −9.374926029770812745189207620591, −8.342269316485315092441089560446, −7.84261529258016309172135468118, −6.43584126441933528374313608425, −5.68280733791041058273165830676, −4.51131880856165925700181116171, −3.08583661968968612764015978118, −0.866087841543724400601761598973, 1.57879843614538794365131137200, 3.19346068723694222992876887276, 4.12592636405168422921552723852, 5.51855206258890072817998252812, 7.22057338555977952145283121708, 7.37209212560432781948332279121, 8.911771334827100226404133927301, 9.792344904414724789176825382712, 10.33639435740983412785178352322, 11.63723860492321696046302496073

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