↑

Properties

Label 2-294-7.4-c3-0-18
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)2-s + (1.5 − 2.59i)3-s + (−1.99 + 3.46i)4-s + (−3 − 5.19i)5-s + 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (6 − 10.3i)10-s + (15 − 25.9i)11-s + (6.00 + 10.3i)12-s − 53·13-s − 18·15-s + (−8 − 13.8i)16-s + (−42 + 72.7i)17-s + (9 − 15.5i)18-s + (−48.5 − 84.0i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.268 − 0.464i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.189 − 0.328i)10-s + (0.411 − 0.712i)11-s + (0.144 + 0.249i)12-s − 1.13·13-s − 0.309·15-s + (−0.125 − 0.216i)16-s + (−0.599 + 1.03i)17-s + (0.117 − 0.204i)18-s + (−0.585 − 1.01i)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

\(L(2)\) \(\approx\) \(1.054889864\)
\(L(\frac12)\) \(\approx\) \(1.054889864\)
\(L(\frac{5}{2})\) 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 \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-15 + 25.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 53T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (48.5 + 84.0i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 180T + 2.43e4T^{2} \)
31 \( 1 + (-89.5 + 155. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-72.5 - 125. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 126T + 6.89e4T^{2} \)
43 \( 1 + 325T + 7.95e4T^{2} \)
47 \( 1 + (183 + 316. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-384 + 665. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (132 - 228. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-409 - 708. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-261.5 + 452. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 342T + 3.57e5T^{2} \)
73 \( 1 + (21.5 - 37.2i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-585.5 - 1.01e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 810T + 5.71e5T^{2} \)
89 \( 1 + (300 + 519. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 386T + 9.12e5T^{2} \)
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

−11.32635511014385337495089751103, −9.948213639733258967646614336823, −8.695430511141415806803508417600, −8.255279729075972079095142687935, −7.02165032012570114678519278852, −6.24330162926711574691797188465, −4.93341377206260618906589441098, −3.86978267469246152517173385039, −2.31897163716872579123307128941, −0.31737253894562537102136508648, 1.97283841003663071407784653991, 3.20729547127759852639819060712, 4.30948530926215371258938081246, 5.27645817280584977826617447898, 6.77653210343503749617671502384, 7.75973394142389535973324429192, 9.173443539561046308474640757567, 9.809996676667652608263405024575, 10.71731277542905959397216823196, 11.64412085671395109896002696873

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