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Properties

Label 2-294-7.4-c1-0-2
Degree $2$
Conductor $294$
Sign $0.386 - 0.922i$
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 2.59i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)10-s + (−1.5 + 2.59i)11-s + (0.499 + 0.866i)12-s + 4·13-s + 3·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−2 − 3.46i)19-s − 3·20-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 + (−0.452 + 0.783i)11-s + (0.144 + 0.249i)12-s + 1.10·13-s + 0.774·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (−0.458 − 0.794i)19-s − 0.670·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47441 + 0.980753i\)
\(L(\frac12)\) \(\approx\) \(1.47441 + 0.980753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−12.14814796756962491231152969332, −10.94663304372568654803690152265, −10.12995107750951951828227135541, −8.927862783067826846508157032946, −7.924628609479120144057344214047, −6.77290324066127448857640223373, −6.41161157954552699571805603448, −5.03695989968728934664911248661, −3.44572890759647428492004949300, −2.23391888154599838654479472433, 1.40264919238267836223954523794, 3.08924076079537029196784210471, 4.35612756688499510285107816568, 5.35613875150297397452357254668, 6.24589082920275981337306364604, 8.355759837900032899939204398645, 8.702230901876412996209469040661, 9.882133240093658466959944347700, 10.55647746576914540557831981482, 11.62728483000516633398829236736

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