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  :  $\chi_{378} (295, \cdot )$
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

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

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