Properties

Label 2-294-7.4-c3-0-8
Degree $2$
Conductor $294$
Sign $0.386 - 0.922i$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (−1.5 + 2.59i)3-s + (−1.99 + 3.46i)4-s + (−7.5 − 12.9i)5-s − 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (15 − 25.9i)10-s + (4.5 − 7.79i)11-s + (−6.00 − 10.3i)12-s + 88·13-s + 45·15-s + (−8 − 13.8i)16-s + (−42 + 72.7i)17-s + (9 − 15.5i)18-s + (52 + 90.0i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.670 − 1.16i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + (0.123 − 0.213i)11-s + (−0.144 − 0.249i)12-s + 1.87·13-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (−0.599 + 1.03i)17-s + (0.117 − 0.204i)18-s + (0.627 + 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.750762449\)
\(L(\frac12)\) \(\approx\) \(1.750762449\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - 1.73i)T \)
3 \( 1 + (1.5 - 2.59i)T \)
7 \( 1 \)
good5 \( 1 + (7.5 + 12.9i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-4.5 + 7.79i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 88T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-52 - 90.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-42 - 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 51T + 2.43e4T^{2} \)
31 \( 1 + (-92.5 + 160. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (22 + 38.1i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 168T + 6.89e4T^{2} \)
43 \( 1 - 326T + 7.95e4T^{2} \)
47 \( 1 + (69 + 119. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (319.5 - 553. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-79.5 + 137. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-361 - 625. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-83 + 143. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.08e3T + 3.57e5T^{2} \)
73 \( 1 + (-109 + 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-291.5 - 504. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 597T + 5.71e5T^{2} \)
89 \( 1 + (519 + 898. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 169T + 9.12e5T^{2} \)
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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

−11.59167692088573204120501530210, −10.75211630743941007668891537802, −9.335125065489385948464744336181, −8.530715602160097581742047109181, −7.87490243641093422137524217029, −6.27720941470193590136274175373, −5.54849671406903208552627925034, −4.26795454360335452991228279177, −3.67990035347657234467470825440, −1.04725889721364366007117534790, 0.830477582989188148630957599919, 2.60408323786689834479359654183, 3.59540763886032649512861377942, 4.93006153973490044115614601044, 6.41390495457370764148171256329, 6.97368471538113584795911811639, 8.260126409059750946534249849781, 9.383109024441158913985252014042, 10.86052495199959519896007248858, 11.06623633803055245897894297780

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