↑

Properties

Label 2-294-7.4-c5-0-5
Degree $2$
Conductor $294$
Sign $-0.900 - 0.435i$
Analytic cond. $47.1528$
Root an. cond. $6.86679$
Motivic weight $5$
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 + 3.46i)2-s + (−4.5 + 7.79i)3-s + (−7.99 + 13.8i)4-s + (−51.7 − 89.6i)5-s − 36·6-s − 63.9·8-s + (−40.5 − 70.1i)9-s + (206. − 358. i)10-s + (−120. + 208. i)11-s + (−72 − 124. i)12-s + 805.·13-s + 931.·15-s + (−128 − 221. i)16-s + (646. − 1.12e3i)17-s + (162 − 280. i)18-s + (137. + 238. i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.925 − 1.60i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.654 − 1.13i)10-s + (−0.299 + 0.518i)11-s + (−0.144 − 0.249i)12-s + 1.32·13-s + 1.06·15-s + (−0.125 − 0.216i)16-s + (0.542 − 0.939i)17-s + (0.117 − 0.204i)18-s + (0.0875 + 0.151i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(47.1528\)
Root analytic conductor: \(6.86679\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :5/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.6746817880\)
\(L(\frac12)\) \(\approx\) \(0.6746817880\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2 - 3.46i)T \)
3 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 \)
good5 \( 1 + (51.7 + 89.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (120. - 208. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 805.T + 3.71e5T^{2} \)
17 \( 1 + (-646. + 1.12e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-137. - 238. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (1.89e3 + 3.28e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 1.22e3T + 2.05e7T^{2} \)
31 \( 1 + (2.81e3 - 4.87e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.53e3 - 7.86e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.82e4T + 1.15e8T^{2} \)
43 \( 1 + 1.17e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.15e4 - 1.99e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (8.83e3 - 1.52e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (9.18e3 - 1.59e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (5.66e3 + 9.80e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.80e4 - 3.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 6.34e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.64e4 + 4.58e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.42e4 - 4.20e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + (5.41e4 + 9.38e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 9.96e4T + 8.58e9T^{2} \)
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

−11.67746599532900493366641035191, −10.36386178977784635372531329472, −9.129144071586550605333348675836, −8.456192258112928217952465554590, −7.63111350276851282839529306859, −6.23335823569125001920137255739, −5.05909528642798878902088862602, −4.50958013114299861744887360528, −3.44172828692050778258738652739, −1.09198641351113775959050559666, 0.19472807488905400749879755380, 1.83577275211195568490860382596, 3.28653292807623554903241064502, 3.81783796757596073178725100313, 5.69018738939597738078411138806, 6.46747154302176764150305047309, 7.58033851108019026727045520910, 8.407207801739574347314267956758, 10.05141424857939280733624960394, 10.86049903970582339848557020601

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