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Properties

Label 2-294-7.2-c3-0-4
Degree $2$
Conductor $294$
Sign $0.900 - 0.435i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−1.94 + 3.37i)5-s − 6·6-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (3.89 + 6.75i)10-s + (30.6 + 53.1i)11-s + (−6.00 + 10.3i)12-s − 53.6·13-s + 11.6·15-s + (−8 + 13.8i)16-s + (16.0 + 27.7i)17-s + (9 + 15.5i)18-s + (−27.8 + 48.3i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.174 + 0.302i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.123 + 0.213i)10-s + (0.841 + 1.45i)11-s + (−0.144 + 0.249i)12-s − 1.14·13-s + 0.201·15-s + (−0.125 + 0.216i)16-s + (0.228 + 0.396i)17-s + (0.117 + 0.204i)18-s + (−0.336 + 0.583i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.900 - 0.435i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.900 - 0.435i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.408896613\)
\(L(\frac12)\) \(\approx\) \(1.408896613\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.94 - 3.37i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.6 - 53.1i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 53.6T + 2.19e3T^{2} \)
17 \( 1 + (-16.0 - 27.7i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (27.8 - 48.3i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-47.3 + 81.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 138.T + 2.43e4T^{2} \)
31 \( 1 + (-66.3 - 114. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (74.6 - 129. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 427.T + 6.89e4T^{2} \)
43 \( 1 - 437.T + 7.95e4T^{2} \)
47 \( 1 + (-28.5 + 49.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-131. - 228. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-225. - 391. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-289. + 501. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (154. + 268. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 1.05e3T + 3.57e5T^{2} \)
73 \( 1 + (-596. - 1.03e3i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (659. - 1.14e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + (116. - 201. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.60e3T + 9.12e5T^{2} \)
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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

−11.68387915724314191730509225463, −10.48158153751445029593357935184, −9.847555465184796782571227715837, −8.623983604249980925815771052059, −7.26378143536653215975711967132, −6.60372996419066880127505198399, −5.16629368037398984647638322641, −4.19239174552200520373212797074, −2.66529141613082991739467892220, −1.41428942819612988460785934131, 0.51754179773313419736121173588, 2.99553128745897793363117424228, 4.23782866927490516846287943716, 5.19940468006627795350918963565, 6.20613057526227723316269783651, 7.23962834385830105442608744828, 8.487317574087692503195168339170, 9.181651237288637384355233238640, 10.31175296453379733410180954257, 11.50728380705404535270388168840

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