Properties

Degree $2$
Conductor $294$
Sign $-0.997 - 0.0633i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (1.5 + 0.866i)3-s + (−0.999 + 1.73i)4-s + (−5.12 + 2.95i)5-s + 2.44i·6-s − 2.82·8-s + (1.5 + 2.59i)9-s + (−7.24 − 4.18i)10-s + (0.878 − 1.52i)11-s + (−2.99 + 1.73i)12-s + 18.7i·13-s − 10.2·15-s + (−2.00 − 3.46i)16-s + (−20.3 − 11.7i)17-s + (−2.12 + 3.67i)18-s + (−19.9 + 11.5i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.5 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−1.02 + 0.591i)5-s + 0.408i·6-s − 0.353·8-s + (0.166 + 0.288i)9-s + (−0.724 − 0.418i)10-s + (0.0798 − 0.138i)11-s + (−0.249 + 0.144i)12-s + 1.44i·13-s − 0.682·15-s + (−0.125 − 0.216i)16-s + (−1.19 − 0.690i)17-s + (−0.117 + 0.204i)18-s + (−1.05 + 0.606i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.997 - 0.0633i$
Motivic weight: \(2\)
Character: $\chi_{294} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ -0.997 - 0.0633i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0398569 + 1.25738i\)
\(L(\frac12)\) \(\approx\) \(0.0398569 + 1.25738i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (-1.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (5.12 - 2.95i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-0.878 + 1.52i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 18.7iT - 169T^{2} \)
17 \( 1 + (20.3 + 11.7i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (19.9 - 11.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (9.36 + 16.2i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 30T + 841T^{2} \)
31 \( 1 + (7.45 + 4.30i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-35.4 - 61.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 41.3iT - 1.68e3T^{2} \)
43 \( 1 - 10.4T + 1.84e3T^{2} \)
47 \( 1 + (-33.5 + 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-18.5 + 32.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-84.4 - 48.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-14.4 + 8.36i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (30.4 - 52.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 110.T + 5.04e3T^{2} \)
73 \( 1 + (-49.1 - 28.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-34.9 - 60.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 6.43iT - 6.88e3T^{2} \)
89 \( 1 + (36.4 - 21.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 51.7iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.87697628492047624680901191863, −11.29706377751817352501183803269, −10.08447283911005827431450693736, −8.862402404848567803043637181799, −8.193330988592548779873497908426, −7.05234682051587544273261600142, −6.39320548256240311305814639110, −4.55537043851791870286105648100, −3.99420022922746849639968796605, −2.56449419282933204546869393149, 0.50554266829983590438363000804, 2.32740498244718668295045756806, 3.72876978288576317481396792878, 4.56979915207340690639701127954, 5.97725140538962698289242091342, 7.38984346726022459837412422355, 8.350372312551556340697704115193, 9.030068745514118689854526263715, 10.39134383748512289814300719610, 11.15144173605365713722016822109

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