Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $-0.547 + 0.836i$
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 + (2.18 − 3.78i)5-s + (−0.5 − 0.866i)7-s + 0.999·8-s − 4.37·10-s + (−0.686 − 1.18i)11-s + (−1 + 1.73i)13-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s − 1.37·17-s + 5·19-s + (2.18 + 3.78i)20-s + (−0.686 + 1.18i)22-s + (−0.813 + 1.40i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.977 − 1.69i)5-s + (−0.188 − 0.327i)7-s + 0.353·8-s − 1.38·10-s + (−0.206 − 0.358i)11-s + (−0.277 + 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s − 0.332·17-s + 1.14·19-s + (0.488 + 0.846i)20-s + (−0.146 + 0.253i)22-s + (−0.169 + 0.293i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 + 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.547 + 0.836i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (253, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ -0.547 + 0.836i)$
$L(1)$  $\approx$  $0.560633 - 1.03733i$
$L(\frac12)$  $\approx$  $0.560633 - 1.03733i$
$L(\frac{3}{2})$   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 \( 1 + (-2.18 + 3.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.37T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.37 + 7.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-2.31 + 4.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.05 - 7.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.74T + 53T^{2} \)
59 \( 1 + (5.05 - 8.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.55 + 2.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + (-2.55 - 4.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.74 - 15.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (-4.05 - 7.02i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06040969111084720439223021798, −9.752984480016948679730192480279, −9.482216234507993353007197704572, −8.551501812870936030503557090636, −7.57753157859405492013002111620, −6.01462663131127110654953456431, −5.07219607525224819657031538209, −4.01935908558319669196968515302, −2.24747080160459599453703670913, −0.925166825961432622441268481514, 2.16959692094955805268421014912, 3.33027456357761475730101904344, 5.26204126986445747167639892258, 6.06518795392006075871071646948, 7.00372585369883507184161938496, 7.62179457492301691383139650658, 9.108673482388802673005288463382, 9.845182513350661738337841704416, 10.54741358872363195700114649960, 11.34841912259692096778283861179

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