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Properties

Label 2-294-7.4-c3-0-16
Degree $2$
Conductor $294$
Sign $0.605 + 0.795i$
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 + (−3 − 5.19i)5-s + 6·6-s − 7.99·8-s + (−4.5 − 7.79i)9-s + (6 − 10.3i)10-s + (−6 + 10.3i)11-s + (6.00 + 10.3i)12-s + 38·13-s − 18·15-s + (−8 − 13.8i)16-s + (63 − 109. i)17-s + (9 − 15.5i)18-s + (−10 − 17.3i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.268 − 0.464i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.189 − 0.328i)10-s + (−0.164 + 0.284i)11-s + (0.144 + 0.249i)12-s + 0.810·13-s − 0.309·15-s + (−0.125 − 0.216i)16-s + (0.898 − 1.55i)17-s + (0.117 − 0.204i)18-s + (−0.120 − 0.209i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(294\)    =    \(2 \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(17.3465\)
Root analytic conductor: \(4.16492\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{294} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 294,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.981567867\)
\(L(\frac12)\) \(\approx\) \(1.981567867\)
\(L(\frac{5}{2})\) 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 \( 1 + (3 + 5.19i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (6 - 10.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 38T + 2.19e3T^{2} \)
17 \( 1 + (-63 + 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (10 + 17.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (84 + 145. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 30T + 2.43e4T^{2} \)
31 \( 1 + (-44 + 76.2i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (127 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 42T + 6.89e4T^{2} \)
43 \( 1 + 52T + 7.95e4T^{2} \)
47 \( 1 + (-48 - 83.1i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (99 - 171. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-330 + 571. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-269 - 465. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (442 - 765. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 792T + 3.57e5T^{2} \)
73 \( 1 + (109 - 188. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 492T + 5.71e5T^{2} \)
89 \( 1 + (405 + 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−11.52250115660518639066056619125, −10.14601432424928239515905997812, −8.974750521434644219870299421603, −8.206813332119895147686359917040, −7.31809189045353135658640291366, −6.33081464851533356057968673043, −5.17247272777067386828451292187, −4.04730999198747845980893691058, −2.62038889676436531280665366353, −0.67223487636697289503251247778, 1.57043150919784029300467850666, 3.26684155493568800381929084241, 3.84569038594634184972540747755, 5.31736457375632353533191954736, 6.31737330122751099221565825940, 7.83492714656439667467744295333, 8.689538352098326201014521451541, 9.898314026878614402867199787749, 10.57896567804243181658801041870, 11.35910906172623415068274700147

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