↑

Properties

Label 2-294-7.2-c1-0-0
Degree $2$
Conductor $294$
Sign $-0.701 - 0.712i$
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

Related objects

Downloads

Learn more

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 + 1.73i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.999 − 1.73i)10-s + (2 + 3.46i)11-s + (−0.499 + 0.866i)12-s − 6·13-s + 1.99·15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (−0.499 − 0.866i)18-s + (−2 + 3.46i)19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (0.603 + 1.04i)11-s + (−0.144 + 0.249i)12-s − 1.66·13-s + 0.516·15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.117 − 0.204i)18-s + (−0.458 + 0.794i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(2.34760\)
Root analytic conductor: \(1.53218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.230379 + 0.549722i\)
\(L(\frac12)\) \(\approx\) \(0.230379 + 0.549722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13088943882806020005347097349, −11.21874463868288977456539625125, −10.08241260517181695135022430656, −9.408518829385937771906293456981, −7.80714630839587868580542486676, −7.40517947442939017957074667149, −6.49004749384355708572760411240, −5.32582021709298821923335436848, −3.93988941373864831048355907815, −2.04072986579305853023600278227, 0.50536055101104460612865375100, 2.71582742669109014304947578532, 4.21536856473896139180963478796, 4.97636167220636002120561114835, 6.47275686945982252098654723090, 7.88157836693051664150520105718, 8.766647804035256629565796305818, 9.571322768419506760571051977178, 10.48893098039013274090130252498, 11.52139013956716966692339485737

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