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Properties

Label 2-297-11.9-c1-0-7
Degree $2$
Conductor $297$
Sign $0.760 + 0.649i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.540 − 1.66i)2-s + (−0.854 − 0.620i)4-s + (1.19 + 3.67i)5-s + (−0.710 − 0.516i)7-s + (1.33 − 0.970i)8-s + 6.74·10-s + (1.60 + 2.90i)11-s + (1.15 − 3.54i)13-s + (−1.24 + 0.902i)14-s + (−1.54 − 4.75i)16-s + (0.595 + 1.83i)17-s + (2.27 − 1.65i)19-s + (1.25 − 3.87i)20-s + (5.69 − 1.09i)22-s + 0.221·23-s + ⋯
L(s)  = 1  + (0.381 − 1.17i)2-s + (−0.427 − 0.310i)4-s + (0.533 + 1.64i)5-s + (−0.268 − 0.195i)7-s + (0.472 − 0.343i)8-s + 2.13·10-s + (0.482 + 0.875i)11-s + (0.319 − 0.984i)13-s + (−0.331 + 0.241i)14-s + (−0.386 − 1.18i)16-s + (0.144 + 0.444i)17-s + (0.522 − 0.379i)19-s + (0.281 − 0.866i)20-s + (1.21 − 0.233i)22-s + 0.0462·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.760 + 0.649i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68381 - 0.621283i\)
\(L(\frac12)\) \(\approx\) \(1.68381 - 0.621283i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-1.60 - 2.90i)T \)
good2 \( 1 + (-0.540 + 1.66i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.19 - 3.67i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.710 + 0.516i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.15 + 3.54i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.595 - 1.83i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.27 + 1.65i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.221T + 23T^{2} \)
29 \( 1 + (6.81 + 4.94i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.95 - 6.01i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.84 + 6.42i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.47 - 3.24i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.57T + 43T^{2} \)
47 \( 1 + (-9.12 + 6.63i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (3.19 - 9.84i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.88 + 1.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.03 - 6.25i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 3.06T + 67T^{2} \)
71 \( 1 + (-0.354 - 1.09i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.953 + 0.692i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.43 + 10.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.55 + 4.78i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 + (0.125 - 0.387i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.55289719584229432995409407878, −10.59101542078274961868940487561, −10.31570709227416686044177642463, −9.325157925846677962605628338609, −7.47816506627305969266407754619, −6.85986211269563204552700386910, −5.58022864240927355784518588028, −3.87205372812526789966148392489, −3.06218564866024013046707620858, −1.92189377112296229626404049989, 1.56621924236009618477825433990, 3.96991910930434107816530380907, 5.17111761179821385792777459301, 5.76846106942281271951528887123, 6.78167859769725291278362414529, 8.030441019766676157885614725599, 8.917918667165010014340529022932, 9.497639163871796572505696232464, 11.07437963689832424954039479046, 12.07968406229733241042252443227

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