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  :  $\chi_{378} (295, \cdot )$
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

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

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