Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $0.711 + 0.703i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 + 0.342i)2-s + (1.71 + 0.214i)3-s + (0.766 − 0.642i)4-s + (−0.701 − 3.97i)5-s + (−1.68 + 0.386i)6-s + (0.766 + 0.642i)7-s + (−0.500 + 0.866i)8-s + (2.90 + 0.737i)9-s + (2.01 + 3.49i)10-s + (−0.324 + 1.84i)11-s + (1.45 − 0.940i)12-s + (0.134 + 0.0489i)13-s + (−0.939 − 0.342i)14-s + (−0.352 − 6.98i)15-s + (0.173 − 0.984i)16-s + (−2.90 − 5.03i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (0.992 + 0.123i)3-s + (0.383 − 0.321i)4-s + (−0.313 − 1.77i)5-s + (−0.689 + 0.157i)6-s + (0.289 + 0.242i)7-s + (−0.176 + 0.306i)8-s + (0.969 + 0.245i)9-s + (0.638 + 1.10i)10-s + (−0.0979 + 0.555i)11-s + (0.419 − 0.271i)12-s + (0.0373 + 0.0135i)13-s + (−0.251 − 0.0914i)14-s + (−0.0909 − 1.80i)15-s + (0.0434 − 0.246i)16-s + (−0.704 − 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.711 + 0.703i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (169, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ 0.711 + 0.703i)$
$L(1)$  $\approx$  $1.24763 - 0.512574i$
$L(\frac12)$  $\approx$  $1.24763 - 0.512574i$
$L(\frac{3}{2})$   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 \( 1 + (0.939 - 0.342i)T \)
3 \( 1 + (-1.71 - 0.214i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
good5 \( 1 + (0.701 + 3.97i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (0.324 - 1.84i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-0.134 - 0.0489i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (2.90 + 5.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.89 + 5.02i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.84 + 3.22i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.917 + 0.333i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.339 + 0.284i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-5.61 - 9.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.19 + 1.89i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.36 - 7.73i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.648 - 0.544i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 1.33T + 53T^{2} \)
59 \( 1 + (-1.11 - 6.29i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (3.40 + 2.85i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (10.4 + 3.81i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-8.19 - 14.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.37 - 4.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.0 + 4.76i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.71 - 2.08i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (7.33 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.20 + 6.85i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.27937778392078358875127359315, −9.806328963766512156251666754824, −9.203404584869898153685057284179, −8.611510854306325328843640998056, −7.84436737148843373296736110850, −6.87847733633078719186843742317, −5.01839005419895750567134193378, −4.56137345841691263352278168078, −2.65760509903550317201465595517, −1.12287684533827572201688538925, 1.93202047470728197306742524262, 3.17072233000105987731955856039, 3.85129595686895411304415708020, 6.11324794771159390392118762527, 7.13116845508673246584666119700, 7.75027657079626296804100141789, 8.607061134731022219856896073135, 9.752873233163737997298741428610, 10.58666797138606984708882840326, 11.08931832809781877083416992453

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