Properties

Degree 2
Conductor $ 2 \cdot 3^{3} \cdot 7 $
Sign $-0.0424 - 0.999i$
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 + (1.65 + 0.505i)3-s + (−0.939 + 0.342i)4-s + (1.31 + 1.10i)5-s + (−0.210 + 1.71i)6-s + (−0.939 − 0.342i)7-s + (−0.5 − 0.866i)8-s + (2.48 + 1.67i)9-s + (−0.856 + 1.48i)10-s + (−0.960 + 0.806i)11-s + (−1.72 + 0.0916i)12-s + (−0.652 + 3.69i)13-s + (0.173 − 0.984i)14-s + (1.61 + 2.48i)15-s + (0.766 − 0.642i)16-s + (3.04 − 5.27i)17-s + ⋯
L(s)  = 1  + (0.122 + 0.696i)2-s + (0.956 + 0.291i)3-s + (−0.469 + 0.171i)4-s + (0.587 + 0.492i)5-s + (−0.0857 + 0.701i)6-s + (−0.355 − 0.129i)7-s + (−0.176 − 0.306i)8-s + (0.829 + 0.558i)9-s + (−0.270 + 0.469i)10-s + (−0.289 + 0.243i)11-s + (−0.499 + 0.0264i)12-s + (−0.180 + 1.02i)13-s + (0.0464 − 0.263i)14-s + (0.417 + 0.642i)15-s + (0.191 − 0.160i)16-s + (0.739 − 1.28i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(378\)    =    \(2 \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.0424 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{378} (337, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 378,\ (\ :1/2),\ -0.0424 - 0.999i)$
$L(1)$  $\approx$  $1.34833 + 1.40688i$
$L(\frac12)$  $\approx$  $1.34833 + 1.40688i$
$L(\frac{3}{2})$   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 \( 1 + (-1.65 - 0.505i)T \)
7 \( 1 + (0.939 + 0.342i)T \)
good5 \( 1 + (-1.31 - 1.10i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (0.960 - 0.806i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.652 - 3.69i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.04 + 5.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 2.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.42 - 0.881i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.744 + 4.22i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.67 - 0.608i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.699 + 1.21i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.568 + 3.22i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.86 + 4.91i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.56 + 2.38i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 3.54T + 53T^{2} \)
59 \( 1 + (6.26 + 5.25i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-10.9 - 3.99i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (2.70 - 15.3i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.68 + 9.84i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.34 + 5.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.215 + 1.22i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.957 + 5.43i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (8.40 + 14.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.1 + 9.32i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.66663720048072117427894067195, −10.17524674656958907882823605809, −9.735994218427490837100077504150, −8.889934675072551780098790455377, −7.68982820954070021684731093724, −7.06844642125495355920680554333, −5.91015084606466115226819904312, −4.67585002599937812937908133273, −3.51221263816703722229489801130, −2.24806431147703686224251074466, 1.35973275009077066035019972678, 2.73065897231044341020928123777, 3.69053266882218623481876545239, 5.15416917949632531457679997948, 6.19934024731004788650528678631, 7.69895679351698126810481005361, 8.464705086654234110410655736238, 9.439606584139590919539821166860, 10.03221274977731523411639450250, 11.02584666561542155228844415077

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