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Properties

Label 2-294-21.2-c2-0-3
Degree $2$
Conductor $294$
Sign $-0.710 - 0.703i$
Analytic cond. $8.01091$
Root an. cond. $2.83035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 0.707i)2-s + (−2.28 − 1.94i)3-s + (0.999 + 1.73i)4-s + (−3.46 − 2i)5-s + (−1.41 − 4i)6-s + 2.82i·8-s + (1.39 + 8.89i)9-s + (−2.82 − 4.89i)10-s + (−8.57 + 4.94i)11-s + (1.09 − 5.89i)12-s + 12.7·13-s + (3.99 + 11.3i)15-s + (−2.00 + 3.46i)16-s + (−27.7 + 16i)17-s + (−4.57 + 11.8i)18-s + (−14.1 + 24.4i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.760 − 0.649i)3-s + (0.249 + 0.433i)4-s + (−0.692 − 0.400i)5-s + (−0.235 − 0.666i)6-s + 0.353i·8-s + (0.155 + 0.987i)9-s + (−0.282 − 0.489i)10-s + (−0.779 + 0.449i)11-s + (0.0913 − 0.491i)12-s + 0.979·13-s + (0.266 + 0.754i)15-s + (−0.125 + 0.216i)16-s + (−1.63 + 0.941i)17-s + (−0.254 + 0.659i)18-s + (−0.744 + 1.28i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.710 - 0.703i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ -0.710 - 0.703i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.238920 + 0.581247i\)
\(L(\frac12)\) \(\approx\) \(0.238920 + 0.581247i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (2.28 + 1.94i)T \)
7 \( 1 \)
good5 \( 1 + (3.46 + 2i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.57 - 4.94i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 12.7T + 169T^{2} \)
17 \( 1 + (27.7 - 16i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (14.1 - 24.4i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (1.22 + 0.707i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 56.5iT - 841T^{2} \)
31 \( 1 + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-25 + 43.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 16iT - 1.68e3T^{2} \)
43 \( 1 + 52T + 1.84e3T^{2} \)
47 \( 1 + (29.4 + 17i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-39.1 + 22.6i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (27.7 - 16i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-16.2 + 28.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 89.0iT - 5.04e3T^{2} \)
73 \( 1 + (-16.2 - 28.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-20 + 34.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 62iT - 6.88e3T^{2} \)
89 \( 1 + (48.4 + 28i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 24.0T + 9.40e3T^{2} \)
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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

−12.14806580092235748310592143522, −11.09934936098290277806887575439, −10.49542296227972767316716492569, −8.582653935885335350137813901133, −7.982615183142407936958230844943, −6.83146573508612491227543696843, −6.01705345354962468845337758559, −4.87861311377826457069545252155, −3.88109801353274499818585533971, −1.90067285717497170323023795147, 0.26496635286619381277892317469, 2.76054541875113861074316028039, 4.03784032468201016511073263040, 4.85316972507921573504561571293, 6.10194975816056495061021600350, 6.93689922047460130150850268398, 8.418987714137010096305915919837, 9.564720994501796107597258070546, 10.70877471391753415001303637648, 11.30974254224430100992041144045

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