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Properties

Label 2-294-3.2-c6-0-33
Degree $2$
Conductor $294$
Sign $0.628 + 0.777i$
Analytic cond. $67.6359$
Root an. cond. $8.22410$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.65i·2-s + (−21 + 16.9i)3-s − 32.0·4-s − 169. i·5-s + (96 + 118. i)6-s + 181. i·8-s + (153. − 712. i)9-s − 960.·10-s + 33.9i·11-s + (672. − 543. i)12-s + 2.95e3·13-s + (2.88e3 + 3.56e3i)15-s + 1.02e3·16-s + 4.48e3i·17-s + (−4.03e3 − 865. i)18-s − 5.25e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.777 + 0.628i)3-s − 0.500·4-s − 1.35i·5-s + (0.444 + 0.549i)6-s + 0.353i·8-s + (0.209 − 0.977i)9-s − 0.960·10-s + 0.0255i·11-s + (0.388 − 0.314i)12-s + 1.34·13-s + (0.853 + 1.05i)15-s + 0.250·16-s + 0.911i·17-s + (−0.691 − 0.148i)18-s − 0.766·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.628 + 0.777i$
Analytic conductor: \(67.6359\)
Root analytic conductor: \(8.22410\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3),\ 0.628 + 0.777i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.392967766\)
\(L(\frac12)\) \(\approx\) \(1.392967766\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 5.65iT \)
3 \( 1 + (21 - 16.9i)T \)
7 \( 1 \)
good5 \( 1 + 169. iT - 1.56e4T^{2} \)
11 \( 1 - 33.9iT - 1.77e6T^{2} \)
13 \( 1 - 2.95e3T + 4.82e6T^{2} \)
17 \( 1 - 4.48e3iT - 2.41e7T^{2} \)
19 \( 1 + 5.25e3T + 4.70e7T^{2} \)
23 \( 1 - 1.02e4iT - 1.48e8T^{2} \)
29 \( 1 + 2.20e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.28e4T + 8.87e8T^{2} \)
37 \( 1 - 3.40e4T + 2.56e9T^{2} \)
41 \( 1 + 1.67e4iT - 4.75e9T^{2} \)
43 \( 1 + 6.40e3T + 6.32e9T^{2} \)
47 \( 1 - 1.79e5iT - 1.07e10T^{2} \)
53 \( 1 - 1.92e5iT - 2.21e10T^{2} \)
59 \( 1 + 3.26e5iT - 4.21e10T^{2} \)
61 \( 1 - 6.25e4T + 5.15e10T^{2} \)
67 \( 1 - 4.38e5T + 9.04e10T^{2} \)
71 \( 1 - 6.82e4iT - 1.28e11T^{2} \)
73 \( 1 - 7.30e5T + 1.51e11T^{2} \)
79 \( 1 - 3.40e5T + 2.43e11T^{2} \)
83 \( 1 - 4.96e5iT - 3.26e11T^{2} \)
89 \( 1 + 3.86e5iT - 4.96e11T^{2} \)
97 \( 1 - 2.81e5T + 8.32e11T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.84877536192690724390954381945, −9.678561776165801439622618395360, −8.964912962212970413795789652548, −8.122551350169719796180131837777, −6.26312424126891370676891768560, −5.41149896765484460196701906779, −4.38123606718352929079478690641, −3.63805800542901853404959116647, −1.59952143282373551629434012210, −0.69289855887798012309923170563, 0.63697940044966219853086672645, 2.27982238035223091395180484751, 3.73310989117289178792387954244, 5.18019736433294002643670552510, 6.32717057836488189052794430638, 6.71662029938284765300737833904, 7.67832725442048462206046043382, 8.715464779560782180618436974184, 10.16978520753493068531050105518, 10.92130337978273030819896555053

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