Properties

Label 2-294-147.101-c1-0-14
Degree $2$
Conductor $294$
Sign $0.249 + 0.968i$
Analytic cond. $2.34760$
Root an. cond. $1.53218$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.930 − 0.365i)2-s + (1.60 − 0.650i)3-s + (0.733 + 0.680i)4-s + (−0.128 − 0.0879i)5-s + (−1.73 + 0.0192i)6-s + (−2.44 − 1.01i)7-s + (−0.433 − 0.900i)8-s + (2.15 − 2.08i)9-s + (0.0879 + 0.128i)10-s + (−0.535 − 3.54i)11-s + (1.61 + 0.614i)12-s + (4.78 − 3.81i)13-s + (1.90 + 1.83i)14-s + (−0.264 − 0.0572i)15-s + (0.0747 + 0.997i)16-s + (1.95 + 0.602i)17-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (0.926 − 0.375i)3-s + (0.366 + 0.340i)4-s + (−0.0576 − 0.0393i)5-s + (−0.707 + 0.00784i)6-s + (−0.923 − 0.382i)7-s + (−0.153 − 0.318i)8-s + (0.717 − 0.696i)9-s + (0.0278 + 0.0407i)10-s + (−0.161 − 1.07i)11-s + (0.467 + 0.177i)12-s + (1.32 − 1.05i)13-s + (0.509 + 0.490i)14-s + (−0.0682 − 0.0147i)15-s + (0.0186 + 0.249i)16-s + (0.473 + 0.146i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.249 + 0.968i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.946626 - 0.734003i\)
\(L(\frac12)\) \(\approx\) \(0.946626 - 0.734003i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.930 + 0.365i)T \)
3 \( 1 + (-1.60 + 0.650i)T \)
7 \( 1 + (2.44 + 1.01i)T \)
good5 \( 1 + (0.128 + 0.0879i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.535 + 3.54i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-4.78 + 3.81i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-1.95 - 0.602i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (2.10 - 1.21i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 8.49i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-4.78 + 1.09i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-1.73 - 1.00i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.28 - 2.11i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (9.06 - 4.36i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (4.79 + 2.30i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (2.30 - 5.86i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-0.683 + 0.736i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-5.97 + 4.07i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-0.373 - 0.403i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (6.76 - 11.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.13 - 1.17i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-1.20 + 0.473i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-8.15 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.06 - 11.3i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (0.649 + 0.0978i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 2.73iT - 97T^{2} \)
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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

−11.49036278848950301840126226187, −10.37355892730515592853548989900, −9.734893003589385280117811528181, −8.466616314735438623589071263904, −8.165454483872630511285639004202, −6.86511506735979496602085604183, −5.91137122629734493765160275724, −3.64428322281689696235531682710, −3.06175695709396295148905501127, −1.10035078193047630491548897608, 2.03299610736499580412360759926, 3.41581763839715301421396832945, 4.75664092335261142210876659410, 6.42300219662143502105750984022, 7.15107476876417607514831463542, 8.506978107157854808152976900677, 8.986472248315944985069884331280, 9.937480817590448197609685828888, 10.62550192847618302728551724838, 11.94603851716426564907527535489

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