Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $0.987 - 0.160i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.766 − 0.642i)2-s + (−1.67 + 0.456i)3-s + (0.173 − 0.984i)4-s + (−1.78 + 0.648i)5-s + (−0.986 + 1.42i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (2.58 − 1.52i)9-s + (−0.947 + 1.64i)10-s + (4.90 + 1.78i)11-s + (0.159 + 1.72i)12-s + (4.25 + 3.57i)13-s + (0.766 + 0.642i)14-s + (2.67 − 1.89i)15-s + (−0.939 − 0.342i)16-s + (2.07 − 3.59i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.964 + 0.263i)3-s + (0.0868 − 0.492i)4-s + (−0.796 + 0.289i)5-s + (−0.402 + 0.581i)6-s + (0.0656 + 0.372i)7-s + (−0.176 − 0.306i)8-s + (0.860 − 0.508i)9-s + (−0.299 + 0.519i)10-s + (1.47 + 0.537i)11-s + (0.0461 + 0.497i)12-s + (1.18 + 0.990i)13-s + (0.204 + 0.171i)14-s + (0.691 − 0.489i)15-s + (−0.234 − 0.0855i)16-s + (0.504 − 0.873i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.987 - 0.160i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (211, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ 0.987 - 0.160i)$
$L(1)$  $\approx$  $1.29709 + 0.104541i$
$L(\frac12)$  $\approx$  $1.29709 + 0.104541i$
$L(\frac{3}{2})$   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 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (1.67 - 0.456i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
good5 \( 1 + (1.78 - 0.648i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 + (-4.90 - 1.78i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-4.25 - 3.57i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.35 - 2.34i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.529 + 3.00i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.78 - 3.17i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (1.79 - 10.1i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-5.29 + 9.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.771 + 0.647i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-8.48 - 3.08i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.04 + 5.90i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + (3.28 - 1.19i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.40 + 13.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-6.83 - 5.73i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.93 - 3.35i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.00 - 8.67i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.44 - 6.24i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-2.87 + 2.41i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (6.98 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.06 + 0.387i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.39046450675757681166959542403, −10.97287851007978031010090438383, −9.670079361554250469296375558317, −8.937155640667549845825739419027, −7.24742088710921904820588389901, −6.51636565913497960740677875275, −5.45796631479852392960911331792, −4.24208468983781471716377898396, −3.59479794811347143620665616144, −1.43556569145556678055849144973, 1.03555920838297663525483149733, 3.65349180541093577790132769535, 4.32088924429220977519137858998, 5.78870474981626470722131101689, 6.26535855157678456401612811057, 7.55364076472082720894159146507, 8.160343934085681784309506722970, 9.486393066379810393171977265505, 10.88351352611071676328957459944, 11.46175788179277097016489658097

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