Properties

Label 2-294-1.1-c3-0-4
Degree $2$
Conductor $294$
Sign $1$
Analytic cond. $17.3465$
Root an. cond. $4.16492$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3·3-s + 4·4-s + 15.8·5-s + 6·6-s − 8·8-s + 9·9-s − 31.7·10-s + 57.3·11-s − 12·12-s − 5.69·13-s − 47.6·15-s + 16·16-s + 51.8·17-s − 18·18-s − 16.2·19-s + 63.5·20-s − 114.·22-s − 213.·23-s + 24·24-s + 127.·25-s + 11.3·26-s − 27·27-s − 218.·29-s + 95.3·30-s + 251.·31-s − 32·32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 1.00·10-s + 1.57·11-s − 0.288·12-s − 0.121·13-s − 0.821·15-s + 0.250·16-s + 0.740·17-s − 0.235·18-s − 0.195·19-s + 0.711·20-s − 1.11·22-s − 1.93·23-s + 0.204·24-s + 1.02·25-s + 0.0859·26-s − 0.192·27-s − 1.39·29-s + 0.580·30-s + 1.45·31-s − 0.176·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.546043252\)
\(L(\frac12)\) \(\approx\) \(1.546043252\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2T \)
3 \( 1 + 3T \)
7 \( 1 \)
good5 \( 1 - 15.8T + 125T^{2} \)
11 \( 1 - 57.3T + 1.33e3T^{2} \)
13 \( 1 + 5.69T + 2.19e3T^{2} \)
17 \( 1 - 51.8T + 4.91e3T^{2} \)
19 \( 1 + 16.2T + 6.85e3T^{2} \)
23 \( 1 + 213.T + 1.21e4T^{2} \)
29 \( 1 + 218.T + 2.43e4T^{2} \)
31 \( 1 - 251.T + 2.97e4T^{2} \)
37 \( 1 - 386.T + 5.06e4T^{2} \)
41 \( 1 - 328.T + 6.89e4T^{2} \)
43 \( 1 + 37.5T + 7.95e4T^{2} \)
47 \( 1 - 254.T + 1.03e5T^{2} \)
53 \( 1 - 211.T + 1.48e5T^{2} \)
59 \( 1 - 412.T + 2.05e5T^{2} \)
61 \( 1 - 836.T + 2.26e5T^{2} \)
67 \( 1 + 165.T + 3.00e5T^{2} \)
71 \( 1 + 465.T + 3.57e5T^{2} \)
73 \( 1 + 449.T + 3.89e5T^{2} \)
79 \( 1 + 343.T + 4.93e5T^{2} \)
83 \( 1 + 1.50e3T + 5.71e5T^{2} \)
89 \( 1 - 341.T + 7.04e5T^{2} \)
97 \( 1 - 865.T + 9.12e5T^{2} \)
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

−11.32060650464516985981258843415, −9.947765621352049508084289630428, −9.817954818624224272394093429000, −8.709411950258291662473298027986, −7.39188270965617798545885839387, −6.17403852664435487822821993804, −5.84128024223951553870130717180, −4.11264196605981104760747501672, −2.20810816843075961671558928849, −1.07004601042491447273294594266, 1.07004601042491447273294594266, 2.20810816843075961671558928849, 4.11264196605981104760747501672, 5.84128024223951553870130717180, 6.17403852664435487822821993804, 7.39188270965617798545885839387, 8.709411950258291662473298027986, 9.817954818624224272394093429000, 9.947765621352049508084289630428, 11.32060650464516985981258843415

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