Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $-0.969 + 0.245i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + (−0.390 + 1.68i)3-s + (−0.939 + 0.342i)4-s + (0.393 + 0.329i)5-s + (−1.72 − 0.0916i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (−2.69 − 1.31i)9-s + (−0.256 + 0.444i)10-s + (−2.36 + 1.98i)11-s + (−0.210 − 1.71i)12-s + (−0.932 + 5.29i)13-s + (0.173 − 0.984i)14-s + (−0.710 + 0.534i)15-s + (0.766 − 0.642i)16-s + (−2.52 + 4.37i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (−0.225 + 0.974i)3-s + (−0.469 + 0.171i)4-s + (0.175 + 0.147i)5-s + (−0.706 − 0.0374i)6-s + (−0.355 − 0.129i)7-s + (−0.176 − 0.306i)8-s + (−0.898 − 0.439i)9-s + (−0.0811 + 0.140i)10-s + (−0.714 + 0.599i)11-s + (−0.0606 − 0.496i)12-s + (−0.258 + 1.46i)13-s + (0.0464 − 0.263i)14-s + (−0.183 + 0.138i)15-s + (0.191 − 0.160i)16-s + (−0.612 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.969 + 0.245i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ -0.969 + 0.245i)$
$L(1)$  $\approx$  $0.103369 - 0.828018i$
$L(\frac12)$  $\approx$  $0.103369 - 0.828018i$
$L(\frac{3}{2})$   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 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (0.390 - 1.68i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
good5 \( 1 + (-0.393 - 0.329i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (2.36 - 1.98i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.932 - 5.29i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.52 - 4.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.98 + 5.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.07 + 1.48i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-0.663 - 3.76i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-7.88 + 2.87i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (3.24 - 5.62i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.73 - 9.83i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (5.49 - 4.61i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-12.7 - 4.63i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 2.76T + 53T^{2} \)
59 \( 1 + (-4.66 - 3.91i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (5.17 + 1.88i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (-0.300 + 1.70i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.87 - 8.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.55 + 8.80i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-1.50 - 8.54i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (0.964 + 1.67i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.3 + 9.56i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.77012770376185357229487762850, −10.75431418488516430273979391405, −9.975131405931042496855395555117, −9.103888619874867956338295466804, −8.305209431984605964589918634725, −6.80043736353107858892681371283, −6.28532245773757685968777533666, −4.77546741806179970922168179829, −4.34534857719467043465416942173, −2.70370745430852939930921421962, 0.53240959774674558491690338395, 2.29588860622682684348711465207, 3.31700702797424029257380303677, 5.18863090811921907563806946024, 5.80640328282853040886654389017, 7.13059579111580261683113943337, 8.128504420678846553840631675589, 8.975780180123411042713595599723, 10.26962840145302206369132337823, 10.87695834758595366614126655534

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