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Properties

Label 2-294-7.3-c2-0-4
Degree $2$
Conductor $294$
Sign $-0.444 - 0.895i$
Analytic cond. $8.01091$
Root an. cond. $2.83035$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 1.22i)2-s + (−1.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s + (1.24 − 0.717i)5-s − 2.44i·6-s − 2.82·8-s + (1.5 + 2.59i)9-s + (1.75 + 1.01i)10-s + (−3 + 5.19i)11-s + (2.99 − 1.73i)12-s + 21.3i·13-s − 2.48·15-s + (−2.00 − 3.46i)16-s + (7.75 + 4.47i)17-s + (−2.12 + 3.67i)18-s + (6.25 − 3.61i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.5 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.248 − 0.143i)5-s − 0.408i·6-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.175 + 0.101i)10-s + (−0.272 + 0.472i)11-s + (0.249 − 0.144i)12-s + 1.64i·13-s − 0.165·15-s + (−0.125 − 0.216i)16-s + (0.456 + 0.263i)17-s + (−0.117 + 0.204i)18-s + (0.329 − 0.190i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(8.01091\)
Root analytic conductor: \(2.83035\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :1),\ -0.444 - 0.895i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.717136 + 1.15592i\)
\(L(\frac12)\) \(\approx\) \(0.717136 + 1.15592i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 1.22i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.24 + 0.717i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 21.3iT - 169T^{2} \)
17 \( 1 + (-7.75 - 4.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.25 + 3.61i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-18.7 - 32.4i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (38.2 + 22.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.9 - 24.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 54.8iT - 1.68e3T^{2} \)
43 \( 1 + 1.48T + 1.84e3T^{2} \)
47 \( 1 + (-37.2 + 21.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-42.7 + 74.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-35.6 - 20.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-1.02 + 0.594i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (2.19 - 3.80i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 137.T + 5.04e3T^{2} \)
73 \( 1 + (68.3 + 39.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (49.1 + 85.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 110. iT - 6.88e3T^{2} \)
89 \( 1 + (-18 + 10.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 10.9iT - 9.40e3T^{2} \)
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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

−11.77800127350047160357580220859, −11.29313761524819345630553019452, −9.757417123415454166244242802876, −9.079111039287400359500495479059, −7.62699993271842239408702110275, −7.00533771215960367820187492418, −5.84841323016575246033415836398, −5.00407783580983836917856320177, −3.74270985401682315727331824666, −1.77946713159777654509790272080, 0.64495859765088003175655096734, 2.64732722040845483727943860177, 3.82535983888995068376404912113, 5.28824307687816261905896741685, 5.81945174018255898279449653017, 7.28304237501173319594435519972, 8.545084482216734082876162786067, 9.682285201599759713164371049357, 10.62553122751861741939370674507, 10.98386818439977750009385973177

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