↑

Properties

Label 2-294-7.6-c4-0-22
Degree $2$
Conductor $294$
Sign $0.409 + 0.912i$
Analytic cond. $30.3907$
Root an. cond. $5.51278$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s + 5.19i·3-s + 8.00·4-s − 12.2i·5-s + 14.6i·6-s + 22.6·8-s − 27·9-s − 34.7i·10-s − 65.1·11-s + 41.5i·12-s − 323. i·13-s + 63.8·15-s + 64.0·16-s + 0.752i·17-s − 76.3·18-s − 313. i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.491i·5-s + 0.408i·6-s + 0.353·8-s − 0.333·9-s − 0.347i·10-s − 0.538·11-s + 0.288i·12-s − 1.91i·13-s + 0.283·15-s + 0.250·16-s + 0.00260i·17-s − 0.235·18-s − 0.867i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(30.3907\)
Root analytic conductor: \(5.51278\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :2),\ 0.409 + 0.912i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.527954323\)
\(L(\frac12)\) \(\approx\) \(2.527954323\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
3 \( 1 - 5.19iT \)
7 \( 1 \)
good5 \( 1 + 12.2iT - 625T^{2} \)
11 \( 1 + 65.1T + 1.46e4T^{2} \)
13 \( 1 + 323. iT - 2.85e4T^{2} \)
17 \( 1 - 0.752iT - 8.35e4T^{2} \)
19 \( 1 + 313. iT - 1.30e5T^{2} \)
23 \( 1 - 293.T + 2.79e5T^{2} \)
29 \( 1 + 83.7T + 7.07e5T^{2} \)
31 \( 1 + 85.5iT - 9.23e5T^{2} \)
37 \( 1 + 2.14e3T + 1.87e6T^{2} \)
41 \( 1 + 1.94e3iT - 2.82e6T^{2} \)
43 \( 1 - 1.06e3T + 3.41e6T^{2} \)
47 \( 1 + 2.77e3iT - 4.87e6T^{2} \)
53 \( 1 - 2.54e3T + 7.89e6T^{2} \)
59 \( 1 - 2.34e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.57e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.12e3T + 2.01e7T^{2} \)
71 \( 1 - 9.25e3T + 2.54e7T^{2} \)
73 \( 1 + 4.41e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.83e3T + 3.89e7T^{2} \)
83 \( 1 - 2.06e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.41e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.65e4iT - 8.85e7T^{2} \)
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

−10.80041822207280978171146237938, −10.36236243740261909606872156527, −9.033833127859160901070272576054, −8.104269097494374855965510555145, −6.95690201822291947675823900183, −5.43910774998746588202123549157, −5.08480741323163460581327006284, −3.66061846708004458685685419887, −2.61312206140534177008091413604, −0.60910315214962633622346881909, 1.56886509509519854142010333766, 2.75338305180999093249396249146, 4.06125959670617210142767075831, 5.29453606647882743083613204366, 6.52844821409879021387790425803, 7.07306537156236594283415113186, 8.253456653001012504698296957298, 9.438596954625833760336554064186, 10.67527677687012967491876408450, 11.45108383151422803920630947894

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